Elementary Statistics Blue Book. The Normal Curve

Transcription

1 Elementary Statistics Blue Book How to work smarter not harder The Normal Curve 68.2% 95.4% 99.7 % Z Scores John G. Blom May 2011

2 01 02 TI 30XA Key Strokes TI 83/84 Key Strokes 08 Vocabulary used in Statistics 09 Symbol Sampler Organizing data, frequency tables & histograms The 5 number summary, Stem & leaf, Box & whisker 17 Mean, median and mode 18 Standard Deviation Using technology Rounding rules, Sample or population, Normal & Chebyshev s rules Probability Probability distributions Binomial distributions The Normal Curve CLT (The central limit theorem) 42 Confidence Intervals large numbers Confidence Intervals small numbers. 46 CI Zc conversion chart & choosing the appropriate distribution Hypothesis testing large numbers & the phone call Hypothesis testing small numbers The linear correlation coefficient for a sample Linear regression equation 62 Chi Square test for independence 63 ± Z tables, t distribution table & Binomial distribution table.

3 TI 30XA Key Strokes 2 nd Key Needs to be pressed to activate the functions above the button keys. 1 PEMDAS Division y x b a c The TI 30XA will do order of arithmetic. Just type in the problem from left to right. When you do multiplication involving a parenthesis you need to type in the times sign between the number and the parenthesis. For example 3( ) must be typed in as 3 x ( ). The numerator (the top number of a fraction) must go into the calculator first. This key raises a base to a power. This is your fraction key. To input into your calculator: Press 3; then press, b a c 4; then press b a c 5. It will be displayed as The b a c key will reduce fractions as well. If you try to reduce a fraction with more than 6 digits (3 digits in the numerator and 3 digits in the denominator) change it to a decimal first. Then use the F D key to change the decimal to a fraction. F D d/c 1 x x 2 x Used to change a decimal to a fraction or vice versa. This will change improper fractions to mixed numbers and vice versa. This is your reciprocal key. (It flips a number.) This key will square a number. The calculator will put the base in parenthesis so be careful. If you want to square 5 you need to do type in 1 5; press the x 2 key; then the equal key; and you will get 25 otherwise you will get +25. This is your square root key or the inverse of squaring a number. x y This is your n th root key. It s the inverse of y x. + This key will change the sign of a number from positive to negative or vice versa. STO Places a number into a memory slot. The memory slots are 1, 2 and 3. RCL Recalls the number you placed in the memory. To clear a memory put 0 into the memory slot. π (pi) Gives you the value of pi (π) to 9 places ( ).

4 2 np r nc r Used for permutations. Put n in first. Then press 2 nd np r. Then put in r and press ENTER. Used for combinations. Put n in first. Then press 2 nd nc r. Then put in r and press ENTER. x! Factorial key. Put in x then press the x! key. Then press ENTER. To find the mean, standard deviation and other good information when you are given raw data or a single class frequency table, turn on the calculator and clear the display. a) Put in the first number. b) Press the + button, you should see n = 1. (Keep doing this until all the numbers are in.) c) Then press 2 nd. Gives you: x The sum of the data points. x N 2 σx n-1 σx n The sample mean. The number of data points. x The sum of the data points squared. The sample standard deviation. The population standard deviation. If you have a frequency distribution table and want to find the mean and standard deviation, turn on the calculator and clear the display. a) Put in the first number. b) Press 2 nd FRQ and put in the frequency. c) Press the + button; you should see n = (whatever the frequency is). (Keep doing this until all the numbers are in.) Then press 2 nd. Gives you: x The sum of the data points. x N 2 The sample mean. The number of data points. x The sum of the data points squared. σx n-1 σx n The sample standard deviation. The population standard deviation.

5 Using the TI 83/84 in Statistics 3 If you want to put raw data in order of size follow these steps: a) Press Stat; then, # 1 Edit. b) Put the data in L 1. c) Press Stat # 2 sort A( d) Press 2 nd L 1 right parenthesis. e) Press Enter. Go back to L 1 and the data is now in order of size. To see the data on the main screen, Press 2 nd Sto, then 2 nd 1 and then enter. Sort A will give you the numbers from smallest to largest. Sort D will give you the numbers from largest to smallest. If you want to clear the data in L 1, press Stat # 4 Clrlist, and then 2 nd L 1. If you want to clear the data in L 1 and L 2, press Stat # 4 Clrlist, and then 2 nd L 1, L 2. If you want the 5 number summary including the mean and standard deviations, s and σ follow these steps: a) Press Stat; then, # 1 Edit. b) Put the data in L 1. When you re done, hit Stat. Cursor over to Calc and press # 1 (1 Var stats). Now press ENTER. Special note: When you use L 1 for x and L 2 for frequency, you need to put the data in L 1 and the frequency in L 2. When you re done hit Stat. Cursor over to Calc and press # 1 (1 Var stats), type in L 1, L 2. Now press ENTER. Binomial Probability Distributions Binomial Probability Distributions Probability Distributions Binomial Exactly a) Press 2 nd Vars. b) Cursor down to Binompdf. c) Press ENTER d) Put in n, p, r in this order. e) Press ENTER. At Most a) Press 2 nd Vars. b) Cursor down to Binomcdf. c) Press ENTER d) Put in n, p, r in this order. r in this case is the at most #. e) Press ENTER. At Least a) 1 Press 2 nd Vars. b) Cursor down to Binomcdf. c) Press ENTER d) Put in n, p, (r 1) in this order. When you do (r 1) use the answer to this arithmetic as your r value and r in this case is the at least #. e) Press ENTER. (At least 8 is the complement of at most 7) Probability functions are found under Math. Then cursor over to PRB. nc r Used for combinations. Put n in first, then press n C r. Then put in r and press ENTER. np r Used for permutations. Put n in first, then press n P r. Then put in r and press ENTER. x! Factorial key. Put in x then press the! key. Now press ENTER. Math 1 enter, will change a decimal to a fraction.

6 To make the normal curve follow these steps: a) Press WINDOW. Set x min to 4. Set x max to +4. Set x scl to 0. Set y min to 0. Set y max to.4. Set y scl to 1. Set x res to 1. Now press y =. Now press 2 nd Vars 1. After the left parenthesis put in the variable x, press the [x,t,,n] key located to the right of the green ALPHA key, and a right parenthesis. Press GRAPH. Z to A: To find the area under the normal curve given one Z score: If you are given a Z score and you are looking for the area under the normal curve to the right of the Z score follow these steps: a) Make the normal curve as shown above. b) Press 2 nd Trace. Press # 7. c) Insert the lower limit which is the given Z score and press ENTER. d) Insert the upper limit which is 4 and press ENTER. The calculator will now shade the curve and give you the area. If you are given a Z score and you are looking for the area under the normal curve to the left of the Z score follow these steps: a) Make the normal curve as shown above. b) Press 2 nd Trace. Press # 7. c) Insert the lower limit which is 4 and press ENTER. d) Insert the upper limit which is the given Z score and press ENTER. The calculator will now shade the curve and give you the area. Special note: 2 nd Prgm 1 will clear the shading of the normal curve If you are doing two tail shading first do the left tail first then the right tail. If you want to find the area between 2 Z scores, follow these steps: Press 2 nd Vars. Press # 2 Normalcdf(. Enter the two Z scores separated by a comma (Left Z score, Right Z score). Press ENTER. 4

7 A Z: If you are given the area under the normal curve and you are looking for the Z score that created this area follow these directions: a) Press 2 nd Vars. b) Press # 3 invnorm. c) Enter the total area to the left of the Z score. d) Press ENTER. If you are given the area under the normal curve between two Z scores, and you are looking for the Z scores that created this area follow these directions: Do the left Z score first then the right: a) Press 2 nd Vars. b) Press # 3 invnorm. c) Enter the total area to the left of the Z score. d) Press ENTER. A to X: If you are given the area under the normal curve and you are looking for the data or X score that created this area follow these directions: a) Press 2 nd Vars. b) Press #3 invnorm. c) Input the data this way: (Total area to the left of the x boundary, the mean, the standard deviation). d) Press ENTER. * If you have a CLT problem, make sure you use the standard error of the mean, σ n when you input the standard deviation into the calculator. If you are given the area under the normal curve and you are looking for two data or x scores that created this area follow these directions: Do the left x score first then the right: a) Press 2 nd Vars. b) Press #3 invnorm. c) Input the data this way: (Total area to the left of the x boundary, the mean, the standard deviation). d) Press ENTER. * If you have a CLT problem, make sure you use the standard error of the mean, σ n when you input the standard deviation into the calculator. 5

8 X to A: If you are given the data or x score and you are looking for the area under the normal curve follow these directions: a) Press 2 nd Vars. b) Press #2 normalcdf. c) Input the data this way: (lower x value, upper x value, the mean, the standard deviation) d) Press ENTER. * If you have no lower limit calculate x at 4 standard deviations from the mean and use that number. This means multiply the standard deviation by 4 then subtract your answer from the mean. ** If you have no upper limit calculate x at 4 standard deviations from the mean and use that number. This means multiply the standard deviation by 4 then add your answer to the mean. *** If you have a CLT problem make sure you use the standard error of the mean when you input the standard deviation into the calculator σ n. C.I. (Confidence Intervals): You can input the data manually (σ, x, n and C.I.) or place the data in L 1. a) Press Stat. Cursor over to Tests. Press # 7 for Z interval or # 8 for t interval. b) Press Data if you are putting the data in L1 otherwise press Stats. c) Fill in the information. d) Press Calculate. Then ENTER The Confidence interval will be in this format (X E, X + E) 6

9 7 Linear Regression equations: First, put the x coordinates in L 1 and the y coordinates in L 2 To find Press xy 2 nd Stat Math 5 sum(l 1 L 2 ) x 2 nd Stat Math 5 sum(l 1 ) y 2 nd Stat Math 5 sum(l 2 ) 2 x 2 nd Stat Math 5 sum(l 1 2 ) [ x] 2 2 nd Stat Math 5 sum(l 1 ) 2 To get m and b, press Stat and cursor over to calc. Press 4 LinReg (ax+b). Now press enter. Another way is to do this is first, put the x coordinates in L 1 and the y coordinates in L 2 Press Vars. Press # 5 Statistics, cursor over to. Select what you want and press ENTER. Special note: the letter r is the correlation coefficient. If you have a TI-83 and after you press Stat, cursor over to Cal and press #4 and you are not getting r do this: a) Press 2 nd catalog. b) Press the x -1 key. c) Scroll down to diagnostic on. d) Press ENTER. e) Press ENTER again (screen should say done). f) Now when you press Stat, cursor over to Cal and press #4 you will get r.

10 Vocabulary Class boundaries Class width Continuous data Convenience sampling Cumulative frequency Cluster sampling Discrete data Frequency Frequency Tables Histogram Mean Median Mode Parameter Probability Distribution Population Qualitative data Quantitative data Random Sample Random Variable Sample Sample Space Sampling Distribution Standard deviation Skewness Statistics Statistic Stratified Sampling Systematic sampling Variance Class boundaries close the gaps produced by the class limits. The difference between two consecutive lower class limits The data have infinitely many possible values, like a measurement of some type are. Data that is easy to obtain. the sum of the frequencies and all previous frequencies. You divide the population into sections or clusters, then randomly select some of those clusters and choose all the members from those selected clusters. data that are countable (finite). the number of times a particular number occurs in the data a table that displays data along with the frequencies of the data a bar graph where the bars are drawn adjacent to each other. a measure of central tendency sometimes called the average. a number that separates the data into two equal parts. the mode is a number found in the data that occurs most frequently. a numerical measurement describing some characteristic of a population Another way to find the mean and standard deviation using probabilities. A population is the complete collection of elements. Qualitative data are nonnumeric. Quantitative data are a numerical value. Each element selected has an equal chance of being selected. A Random Variable is a function, which assigns unique numerical values to all possible outcomes of a random experiment under fixed conditions. a subset of the population A set or collection of outcomes of a particular experiment. When you take repeated samples from a population and keep track of the sample means. the measure of statistical dispersion, measuring how widely spread the data values are. Is a measure of the asymmetry of a distribution. Statistics is a discipline of collection, analysis, interpretation and presentation of data. a numerical measurement describing some characteristic of a sample You subdivide the population into at least two different subgroups that share the same characteristics (such as age or gender) and then draw a sample from each. You select a starting point and select every other element or every 3 rd or 4 th or 5 th. The variance and the standard deviation are measures of how spread out a distribution is. The variance is the square of the standard deviation. 8

11 9 Symbol Sampler Symbol Pronounced Meaning f f frequency Capital Sigma summation x x the data points x Summation x sum of the x values x 2 Summation of x squared sum of the squares of the x values x Summation x square the sum of the x values squared n n number of x values x x bar the sample mean µ Mu the population mean s s the sample standard deviation σ Lower case sigma the population standard deviation s 2 s squared the sample variance σ 2 Sigma squared the population variance Z Z standard score Z c Z c Z score at critical t t student t distribution t c t c t score at critical d.f. d.f. degree of freedom np r np r number of permutations of n items selected r at a time nc r nc r number of combinations of n items selected r at a time μ x Mu of x bar mean of the means. x Sigma of x bar standard error of the mean x = σ n x x title the median α Alpha the total area outside a confidence interval

12 10 Organizing Data: Frequency Tables: a table that displays data along with frequencies Frequency: The number of times a particular number occurs in the data Cumulative frequency: The sum of the frequencies and all previous frequencies. To compute the class width: (The range) (desired number of classes) Round the quotient up to the next whole number if you have a zero remainder. To compute the midpoint: (lowest class limit + upper class limit) 2 To compute the class boundaries: (The lower class limit of the 2 nd class The upper class limit of the 1 st class) 2 Add this amount to each upper class and subtract this amount from each lower class. Class midpoint: The mean of each class. Class width: The difference between two consecutive lower class limits

13 How to construct a frequency distribution table. 11 a) Decide on the number of classes. As a rule of thumb the number of classes should be between 5 and 10, for very large data sets the number of classes will be larger but usually not more than twenty (20). b) Class width = (The largest data value) (The smallest data value) The number of classes Round your answer up to the next whole number even if the remainder is 0. The class width is the quotient plus one more. c) Now chose a number for the lower limit of your table. Use the smallest data value unless you have decimals then use the next lowest whole number. Example if the smallest number in the data was 34.7 you should use 34 as your lower limit. d) Using your lower limit, found in part c, add the class width to this and list all the lower limits. e) Now put in the upper limit of the first class which is one less than the lower limit of the second class. Add the class width to this and list all the upper limits. f) Go back to the data and using tally marks put in the frequency for the appropriate class. g) Now you can put in the class boundaries. Class boundaries are the number used to separate the classes but without the gaps produced by the class limits. To calculate the class boundaries: (The lower class limit of the 2 nd class The upper class limit of the 1 st class) 2 Add this amount to each upper class limit and subtract this amount from each lower class limit. h) The class midpoint can be found by taking the mean of each class. i) The relative frequency = (The class frequency) (The sum of all the frequencies) Because the sum of the relative frequency must equal 1, do not round the relative frequencies, and be careful of terminating and repeating decimals.

14 12 Frequency Histograms Often it is useful to display grouped data as a frequency histogram because a picture is worth a 1,000 words. Histogram: A bar graph where the bars are drawn adjacent to each other Frequency Polygon: A line that connects the top midpoint of each bar on a histogram. Relative Frequency histogram: Same as a histogram except the vertical axis is in terms of percentages instead of frequencies. How to construct a frequency histogram from a frequency distribution table. a) Draw a vertical and horizontal axis and label the grid, the vertical axis is the frequency and the horizontal axis is the class limits or boundaries. b) If the first interval does not start at 0 use a break symbol on the axis. c) Draw bars with heights corresponding to the frequency values in the table. d) Give the graph an appropriate title.

15 13 The 5 number summery Q (quartiles) will divide the data into quarters Q 1 the median of the lower half (25%) Q 2 the median or half the data (50%) Q 3 the median of the upper half (75%) The 5 number summery {smallest number, Q 1, Q 2, Q 3, largest number} Interquartile range (IQR) Q 3 Q 1 What is an outlier? An outlier is a value that is located very far away from all of the other values, an extreme value. An outlier can have a dramatic effect on the mean and standard deviation. How to identify Potential Outliers any data value less than Q 1 [(1.5)(IQR)] or higher than Q 3 + [(1.5)(IQR)] To create the 5 number summary a) Take the given numbers and put them in order of size. b) Now find the number in the middle. This number is called the median or Q 2 and Q 2 divides the numbers into two groups, the bottom half and the top half. c) Now find the median of the bottom half, this number is Q 1 and Q 1 divides the bottom half in two groups, the lower quarter and the upper 75% d) Now find the median of the top half. This number is Q 3 and Q 3 divides the top half in two groups, the upper quarter and the lower 75% e) The smallest number is called Min. and the largest number is called Max.

16 How to find Q 2 if you have an even number of data points: Put the numbers in order of size, count to see how many data points you have and divide this number by 2. Starting with the smallest number count the data points until you get to the quotient and quotient plus one. Find the mean of these two numbers and you have Q 2. Put the numbers in order of size. 2, 4, 6, 8, 9, 12, 34, 45, 67, How many numbers do you have? Divide 10 by 2 10 numbers Count to the 5 th and 6 th number. Q 2 is the mean between the 5 and 6 data point. 2, 4, 6, 8, 9 Q 2 12, 34, 45, 67, 78 Q 2 = = How to find Q 2 if you have an odd number of data points: Put the numbers in order of size, count to see how many data points you have and divide this number by 2. Starting with the smallest number count the data points until you get to the quotient plus one more. Put the numbers in order of size. 2, 4, 6, 8, 9 How many numbers do you have? Divide 5 by 2 5 numbers 2 25 To find Q 2 count one more data point than the quotient. 2, 4, 6, 8, 9 Q 2

17 15 A Stem and Leaf is a display that shows each data value. A stem and leaf is a cute way to put numbers in order of size. The stem is the left digit, but there is no rule as to the number of digits the stem can contain but as a rule of thumb the left digit is the stem and the leaf is everything else. Procedure: a) Find the smallest and largest number b) To the left of a vertical line write each consecutive digit from the smallest stem to the largest stem. c) Insert the leaves. d) Put the leaves in order of size. e) Include a key. A Box and whisker provides a graphical display of the data To create a box and whisker plot. a) Make a number line using a convenient scale starting with the smallest number and ending with the largest number. b) Draw a rectangular box from Q 1 to Q 3. c) In this box place a vertical line at Q 2. d) From Q 1 and Q 3 draw a horizontal line to the smallest and largest number. This is called the whisker

18 16 Hi! I m Marissa and one of the things I like to do is work smarter rather than harder. If you have a TI - 83/84 calculators, smile. Did you know you can put numbers in order of size and find the five number summary on your calculator? Let me show you. If you are given raw data and you want to put data in order of size follow these steps: a) Press Stat. Then # 1 Edit. b) Put the data in L 1. When you put the data in L 1 you need to press ENTER after each number. c) Press Stat. Then # 2 sort A( d) Press 2 nd L 1 right parenthesis. L 1 can be found above the number 1. e) Press ENTER. f) Go back to L 1 and the data is now in order of size. To see the data on the main screen: Press 2 nd Sto. Then 2 nd 1. Then ENTER. Sort A will give you the numbers from smallest to largest. Sort D will give you the numbers from largest to smallest. If you want the 5 number summary follow these steps: a) Press Stat then # 1 Edit b) Put the data in L 1 c) When you re done, Press Stat. Cursor over to Calc and Press # 1 (1 Var stats). d) Now press ENTER. Cursor down and you will find the 5 number summary. If you want to clear the data in L 1, press Stat # 4 Clrlist, and then 2 nd L 1. If you want to clear the data in L 1 and L 2, press Stat # 4 Clrlist, and then 2 nd L 1, L 2. If you do not have L 1 press STAT, then #5, then ENTER.

19 17 Mean, median, mode and range Mean: Add up the numbers and divide by the total number of numbers. Median: The middle number (but you have to arrange the data in order of size) If the number of values is odd.. The median is located in the exact middle. If the number of values is even. The median is found by computing the mean of the two middle numbers. Mode: The value that occurs the most frequently in the given data. (You can have no mode or one or two..) Range: The difference between the largest number and the smallest number in the data. Midrange: (largest number + smallest number) divided by 2 Symbol Symbol μ Symbol x Symbol x Pronounced Sigma and means the sum of the following numbers Pronounced Mu and is used in population mean The letter x is the data value Pronounced x bar and is used in sample mean Formulas: Sample mean x= x n Population mean μ= x N Weighted mean w x x= w Where w is the weight of the data value x Mean (Frequency table) f x x= f Where f is the frequency of the data value x

20 Standard Deviation 18 The standard deviation is the most common measure of statistical dispersion, measuring how widely spread the data values are. If the data points are close to the mean the standard deviation is small and the normal curve will be steep and narrow. If the data points are spread far apart from the mean then the standard deviation is large and the normal curve is relative flat and wide. How to find the Standard Deviation Symbol σ Pronounced Sigma and is used in population standard deviation Population standard deviation σ = (x - x) 2 n Symbol s Pronounced s and is used in sample standard deviation Formula 1: Sample standard deviation (Raw Data) s = [n( x 2)-( x) 2] [n(n-1)] This is the easier of the two formulas Formula 2: Sample standard deviation (Raw Data) s = (x - x) 2 (n - 1) Sample standard deviation (Frequency Table) s = [n( (f *x 2)-( f *x) 2] [n(n-1)] An approximation to the standard deviation is the range rule of thumb s = Range 4 Remember s = the sample standard deviation A population is the complete collection of elements and a sample is a subset of the population. The Coefficient of Variation expresses the SD as a % of the mean. s 100 x The population variance σ 2 Sample variance s 2

21 19 Marissa and Robbie s phone call on do I find the sample or population standard deviation. Hi, Robbie. I was given raw data and the question asked me to calculate the standard deviation. Do I use the sample or population standard deviation? Hi, Marissa. You find the sample standard deviation unless you are told otherwise. Wait! Before you go, what is this range rule of thumb? It is an approximation of the standard deviation. All you do is divide the range, which is the largest number minus the smallest number, by 4. This is not an exact answer, but you can use it to check your calculated answer.

22 20 Hi! I m Robbie. Let me show you how to get the mean and median on your TI 83/84 calculators. If you are given raw data and you want to find the mean and/or the median follow these steps. a) Press Stat then # 1 Edit. b) Put the data in L 1. When you put the data in L 1 you need to press ENTER after each number. c) When you re done hit Stat. Cursor over to Calc and Press # 1 (1 Var stats). d) Now press ENTER. xis the mean. Now cursor down and you will find the median listed as Med. If you have a single class frequency table where you have one column for x and another column for the frequency and you want to find the mean and or the median follow these steps. a) Press Stat then # 1 Edit. b) Put the x data in L 1 and the frequency into L 2. c) When you re done hit Stat. Cursor over to Calc and Press # 1 (1 Var stats). d) Now you need to press 2 nd L 1, 2nd L 2. e) Now press ENTER. xis the sample mean. Now cursor down and you will find the median listed as Med. If you have a frequency distribution table where you have upper and lower class limits and you want to find the mean and or the median follow these steps. a) Find the class midpoint for each class this will serve as your x which goes in L 1 and the frequency will go in L 2. b) Press Stat then # 1 Edit. c) Put x (the class midpoint) in L 1 and the frequency into L 2. d) When you re done press Stat. Cursor over to Calc and Press # 1 (1 Var stats). e) Now you need to press 2 nd L 1, 2 nd L 2. f) Now press ENTER. xis the sample mean. Now cursor down and you will find the median listed as Med.

23 21 Hi! Let me show you how to get the sample or population standard deviation on your TI 83/84 calculators. If you are given raw data and want to find the standard deviation follow these steps: a) Press Stat; then, # 1 Edit. b) Put the data in L 1. When you put the data in L 1 you need to press ENTER after each number. c) When you re done hit Stat. Cursor over to Calc and Press # 1 (1 Var stats). d) Now press ENTER. S is your sample standard deviation and σ is your population standard deviation. If you have a single class frequency table where you have one column for x and another column for the frequency and want to find the standard deviation follow these steps: a) Press Stat; then, # 1 Edit. b) Put the x the data in L 1 and the frequency into L 2. When you put the data in L 1 and L 2 you need to press enter after each number. c) When you re done hit Stat, cursor over to Calc and Press # 1 (1 Var stats). d) Now you need to do press 2 nd L 1, 2 nd L 2. e) Now press ENTER. S is your sample standard deviation and σ is your population standard deviation. If you have a frequency distribution table where you have upper and lower class limits follow these steps: First find the class midpoint for each class; This will serve as your x, which goes in L 1. The frequency will go in L 2. Now you can find the standard deviation. a) Press Stat; then, # 1 Edit. b) Put x the (the class midpoints) in L 1 and the frequency into L 2. When you put the data in L 1 and L 2 you need to press enter after each number. c) When you re done hit Stat. Cursor over to Calc and Press # 1 (1 Var stats). d) Now you need to press 2 nd L 1, 2 nd L 2. e) Now press ENTER. S is your sample standard deviation and σ is your population standard deviation.

24 22 General information used in Statistics Rounding rule: When and where do you round When you have an answer with a lot of decimals in it you should round to the data using normal rounding rules unless you are told otherwise. You should round a Z score to the hundredths. Sample or population? A population is the complete collection of elements and a sample is a subset of the population. Sigma σ is used in a population standard deviation and s is a sample standard deviation. If you are given raw data and are asked to calculate the standard deviation unless you are told to find the population standard deviation, find the sample standard deviation. In confidence intervals and hypothesis testing Sigma σ the population standard deviation is important and must be given in the problem otherwise the population standard deviation is not known. Empirical (Normal) rule: 1 SD = 68.2 % 2 SD = 95.4 % 3 SD = 99.7% 1 Chebyshev s rule 1- k 2 k = number of standard deviations and k > 1 2 SD = 75% 3 SD = 88.9% 4 SD = 93.8%

25 23 Marissa and Robbie s phone call on Chebysyhev s rule. Hi, Robbie. What is Chebyshev's rule? Hi, Marissa. Chebyshev's rule applies to any sample of measurement regardless of the shape of its distribution. The rule states that none of the measurements will fall within one standard deviation of the mean. At least 75% of the data will fall within two standard deviations of the mean, and 88.9% of the data will fall within three standard deviations of the mean, and so on. Are you telling me this is a variation of the normal rule? Yes! Then what is this crazy looking 1 formula: 1- K 2 This is Chebyshev's formula and it is used to calculate the area under the curve when you know the number of standard deviations K. Substitute the value of K (the number of standard deviations) into the formula. Do some arithmetic and you have the area under the curve. That s how they get the numbers shown above. It s a variation of the normal rule. Thanks, Robbie. Bye. Bye, Marissa.

26 Probability information (The key words are underlined and are in Italic) 24 Probability of an event formula Number of outcomes favorable to A P(A) = Total number of outcomes The complement of A If A is an event within the sample space S of an activity or experiment, the complement of A (denoted A ) consists of all outcomes in S that are not in A. The complement of A is everything else in the problem that is NOT in A. The probability of the complement of an event is one minus the probability of the event. P( A ) = 1 - P(A) The complement of A is A (A bar) and means not A. P(A) + P( A ) = 1 To calculate the probability of at least 1 find the probability of none and subtract your answer from 1. Odds in favor of A occurring = P(A) P( A ) Odds against A occurring = P( A ) P(A) P(A or B) = P(A) + P(B) P(A and B) P(A and B) = P(A) P(B) Two events that have NO outcomes in common are called mutually exclusive these are events that cannot occur at the same time. If A and B are mutually exclusive events then. P(A or B) = P(A) + P(B) If events A and B are NOT mutually exclusive then. P(A or B) = P(A) + P(B) - P(A and B)

27 25! is called factorial and 5! = = 120 Factorial Notation is used when you need to work with a series of factors, each being one less than the previous factor. Instead of writing out , we use factorial notation and write 5! = 120. Note: 0! = 1 and 1! = 1 Examples: a) 6! = = 720 b) (10-6)! = 4! = = 24 The Counting Principle is used to find out the total number of ways an event can occur. To use the counting principle multiply the number of ways each activity can occur. If the first event can occur in m ways and the second event can occur in n ways then the event together can occur in m n ways Example: The ice cream shop offers 31 flavors. You order a double-scoop cone. In how many different ways can the clerk put the ice cream on the cone if you wanted two different flavors? Answer 31 x 30 = 930 Permutation and Combinations Permutation Formula n P r = n! (n-r)! Use n P r if you see the words {Arrangements, Permutations, order is important here} Permutation rule when items are identical like letters or colors. n is the number of letters and r is the number of repeated letters n! r! Combination Formula n C r = n! (n-r)!r! Use n C r if you see the words {Collections, Combinations, Committees or selections}

28 Sample Space Single Event 26 Activity Rolling a die Sample Space {1, 2, 3, 4, 5, 6} Tossing a coin { Heads, Tails} Drawing a marble from the jar List the different colored marbles that are in the jar. In this case {blue, red, purple, green and orange} Deck of cards There are 52 cards in a deck. There are four suits, clubs, diamonds, spades and hearts. Each suit has 13 cards, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, queen, king and ace. The Jack, queen and king are the picture cards. Spinner Whatever is on the spinner in this case 1, 2, 3, 4 4 3

29 27 Shown below is a sample space for two activities or experiment. Activity Sample Space Rolling a pair of dice {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} Tossing a coin and rolling a die {(H,1) (H,2) (H,3) (H,4) (H,5) (H,6) (T,1) (T,2) (T,3) (T,4) (T,5) (T,6)} Shown below is a sample space for three activities or experiment. Activity Tossing a coin 3 times Sample Space (HHH) (HHT) (HTH) (HTT) (THH) (THT) (TTH) (TTT)

30 28 Whenever you have multiple events, making the sample space is important you can use a tree or what I call The Box Diagram. Tree diagrams or box diagrams help display the outcomes of an experiment consisting of a series of events Let s make the box diagram for tossing a coin and rolling a die a) List the outcomes for each event. Tossing a coin Rolling a die {Heads, Tails} {1, 2, 3, 4, 5, 6} b) Make a square or rectangular box. c) Put the outcomes for the first event on the outside vertical axis. d) Put the outcomes for the second event on the outside top of the horizontal axis. e) Fill in the grid H H/1 H/2 H/3 H/4 H/5 H/6 T T/1 T/2 T/3 T/4 T/5 T/6

31 29 Hi. Let me show you where you can find your probability function keys on the TI 83/84 calculators. PROBABILITY FUNCTION KEYS are found under Math. Then cursor over to PRB. nc r Used for combinations. Put n in first. Then press the n C r key. Then put in r. Press ENTER. np r Used for permutations. Put n in first. Then press the n P r key. Then put in r. Press ENTER. x! Factorial key. Put x in first. Then press the! key. Then press ENTER.

32 30 Probability Distribution A Probability Distribution is table that shows you all possible outcomes of the random variable x along with the probability of x. If you use the relative frequency you have created a Probability Distribution To determine if you have a valid probability distribution both of the following must be true. P(x) = 1 and 0 P(x) 1 Formulas: μ = [x][p(x)] σ = 2 2 [x P(x)]-μ This is the easier of the two formulas or σ = 2 [x-μ] [P(x)] σ 2 equals the variance The expected value E which is also known as the mean is symbolized as E E = μ = [x][p(x)]

33 31 Hi. Let me show you how easy the new topic called probability distributions will be to do. First, make sure it is a valid probability distribution by looking at P(x). Each probability must be between 0 and 1 inclusively, and the sum of P(x) must equal 1. If you have a probability distribution table, it will list all the x values in a column or row with P(x) next to it. If you want to find the population mean (µ) multiply x and P(x) and add up that column. This total is symbolically written as [x] [P(x)]. To find the population standard deviation ( ) first square x. Then multiply x 2 times P(x) and add up this column. This sum is symbolically written as x 2 P(x). Now from this answer, subtract the square of the mean. Then find the square root of the new sum and you have your answer.

34 32 Binomial Probability Distributions or Bernoulli Experiments. Conditions for Binomial Probability Distributions There are n identical trials Each trial has only two outcomes (success and failure) The probabilities of the two outcomes remain constant for each trial and the trials are independent Notations for Binomial Probability Distributions n = The number of trials x = The number of success in n trials (used in the table) r = The number of success in n trials (used in the formula) p = The probability of getting x success among n trials q = The probability of failure Binomial Probability Distribution Formula P(x) = C P r Q (n-r) n r When do I use the binomial probability distribution??? When you see the words Exactly ; At least ; At most or have repeated trials with the same outcomes. Exactly r {r} At least r {r, r + 1, r + 2, r + 3 n successes} r to n At most r {r, r 1, r - 2, r successes} r to 0 Using probability distributions table n (The number of trials) x (The number of success) and p (Probability of success) First find n and x then align that row with p and you have the answer for P(x) Using your TI 83/84 calculator see page 3 of the blue book for directions. The mean and standard deviation for Binomial Distributions μ = n(p) and σ = n p q Typical values = µ ± 2σ

35 33 In this unit you will study the binomial probability distribution. A binomial probability distribution has the following properties. a) You will have n repeated trials. b) Each trial having two outcomes: Success or failure. c) The probability of success is the same for each and every trial; and the outcome for one trial does not affect the outcome of the other, trials. There are four ways to solve these problems: 1) You can make a sample space. 2) You can use the TI-83/84 calculators. 3) You can use the Binomial formula. 4) You can use the binomial distribution table.

36 34 Hi. Let me show you how to use your TI 83/84 calculators when you have a binomial probability distribution. The key words here are Exactly ; At Most and At Least Exactly a) Press 2 nd Vars. b) Cursor down to Binompdf. c) Press ENTER d) Put in n, p, r in this order. e) Press ENTER. Find the key words to the left and follow the directions in that box. At Most a) Press 2 nd Vars. b) Cursor down to Binomcdf. c) Press ENTER d) Put in n, p, r in this order. r in this case is the at most #. e) Press ENTER. At Least a) 1 Press 2 nd Vars. b) Cursor down to Binomcdf. c) Press ENTER d) Put in n, p, (r 1) in this order. When you do (r 1) use the answer to this arithmetic as your r value and r in this case is the at least #. e) Press ENTER. (At least 8 is the complement of at most 7)

37 35 Hi. Let me show you how to use your binomial distribution table. I always write n, p and r on my paper first. Now go to the binomial distribution table and in the left column find n. Next to n is x, which is the same as r. Move your finger to the number that represents x. Good! Now move your finger across the row until you get to the probability. Below is an important example for you to study. I will use n = 7, a probability of 50%, and the words exactly 4, at most 4 and at least 4. Let s say the question used the word Exactly 4 I would write n = 7, r = 4 and P =.5. Go to the binomial distribution table and in the left column under n find 7. Next to n is x, which is the same as r. Move your finger to 4 and then move your finger across the row until you get to the probability.5. You will find your answer:.2734 Let s say the question used the word At Most 4. I would write n = 7, r = 0, 1, 2, 3, 4 and P =.5 Go to the binomial distribution table and in the left column under n find 7. Next to n is x, which is the same as r. Move your finger to 4 and now move your finger across the row until you get to the probability.5. You will find the answer: Now, do it again for r = 3; then 2; then 1; and then 0. Yes, you will get 5 answers. All you have to do now is add them together for the final answer of Let s say the question used the word At Least 4. I would write n = 7, r = 4, 5, 6, 7 and P =.5 Go to the binomial distribution table and in the left column find 7. Next to n is x, which is the same as r. Move your finger to 4 and now move your finger across the row until you get to the probability.5. You will find the answer: Now, do it again for r = 5; then 6; and then 7. Yes you will get 4 answers. All you have to do now is add them together for the final answer of.5.

38 Properties of the normal curve The total area under the normal curve is The normal curve is symmetric about μ and the area on each side of the mean is ½. 3. The tail of the normal curve approaches the axis asymptotically and extends indefinitely. 4. Each pair of values for μ and σ determines a different curve. Small standard deviations make the curve tall and narrow while large standard deviations make the curve wide and flatter. 5. The highest part of the curve occurs at the mean and can be any real number. 6. The mean, median and mode are all equal in a normal curve. 7. At 1 standard deviation the normal curve go concavity (changes direction) 8. There is a correspondence between the area under the normal curve and probability. The normal curve Unusual values: Any Z value less than 2 or larger than +2

39 Using your Z table 37 1) The Z table is designed only for standard normal distributions which have a mean of 0 and a standard deviation of 1. 2) The value in the table is bounded on the left by a vertical line above a specific Z value. 3) The Z score in the table is the leftmost column and the top row is hundredths. 4) Z scores are sometimes called standard scores. To find the area under the normal distribution curve given Z. {Z A} a) Make a sketch of the normal distribution curve. b) Under the curve draw a Z axis and label the highest point of the curve 0. c) Put in the Z score(s) and shade the given area. d) Use the Z table to find the area to the left of Z score. e) Remember the total area under the curve is 1 and half the area is.5 f) The unit s digit and tenth s digit of the Z score in on the left vertical column g) Across the top of the table is the hundredths digit. Working Backwards: (A Z) Working Backwards: (A Z) given the area under the normal curve and you are asked to find the Z score. Look for the area in the table, you need to get as close as possible to the area so look to the left and to the right then pick the closest one. If your area is in exact middle of two numbers find the mean of the two Z scores which will give you a Z score with a thousandths digit. How to find the area under the normal distribution curve given X {X Z A} a) Make a sketch of the normal distribution curve. b) Under the Z axis draw an x axis. (The x axis is the data axis). Label the highest point of the curve µ c) Change X to Z or Z to X using the formula below How to find a Z score given data (X Z) How to find X (a data point) given Z (Z X) Z = x-μ σ X = [Z σ] + μ d) Now use your Z table to find the area.

40 38 Hi. Let me show you how to get the normal curve, using your TI 83/84 calculators. The first thing you need to do is set up the graphing window on the calculator. On the very top you will see an oval button marked window press it. Set X min to 4, X max to + 4, X scl to 0, Y min to 0, Y max to.4, Y scl to 1and X res to 1. Very good now press 2 nd quit. On the very top of your calculator you will see an oval button marked y = ; press it. Now press 2 nd Vars 1. After the parenthesis put in an x (To put an x after the parenthesis, press the oval button located in the 2 nd column, 2 nd row. It is marked (x,t,o,n). Now close the parenthesis, and press graph which is located at the top right of your calculator. You should now be looking at the normal curve. You are now ready to get the area under the normal curve, given one Z score. To get the area to the right of a Z score follow these steps: Follow the directions given above and get the normal curve. Then press 2 nd, Trace, then 7. The calculator is now asking you for the lower limit; type in the given Z score and press ENTER. The calculator is now asking you for the upper limit; insert the upper limit which is 4 and press ENTER. The calculator will shade in the area under the curve and give you the answer. To get the area to the left of a Z score follow these steps: Follow the directions given above and get the normal curve. Then press 2 nd, Trace, then 7. The calculator is now asking you for the lower limit; type in 4 and press ENTER. The calculator is now asking you for the upper limit; type in the given Z score and press ENTER. The calculator will shade in the area under the curve and give you the answer. This sounds confusing but it s not. If the shading is to the right of the Z score your lower and upper limits will go from the Z score to +4. If the shading is to the left of the Z score your lower and upper limits will go from 4 to the Z score. With a little practice, you will be fine. The 2 nd program 1 button will clear the drawing for you. Press graph and your ready for the next problem. If you are doing two tail shading, do the left tail first, then, the right tail.

41 39 Let me show you how to get the area under the normal curve between two Z scores, using your TI 83/84 calculators. Finding A given two Z scores. Press 2 nd Vars. Press # 2 Normalcdf( Enter the two Z scores separated by a comma. (Left Z score, Right Z score). Press ENTER. Let me show you how to get the Z score given the area under the normal curve, using your TI 83/84 calculators. Press 2 nd Vars. Press # 3 invnorm. Enter the total area to the left of the Z score. Press ENTER. Let me show you how to get the area under the normal curve between two x values using your TI 83/84 calculators. Press 2 nd Vars, then, #2 normalcdf. Input the data this way (lower x value, upper x value, the mean, the standard deviation). Press ENTER. * If you have no lower or upper limit, calculate the lower and upper value at 4 standard deviations from the mean (µ) and use that number. Do this: Multiply the standard deviation by 4 then subtract this number from the mean (µ) to get the lower value. Multiply the standard deviation by 4 then add this number to the mean (µ) to get the upper value. Let me show you how to get X, given the area under the normal curve, using your TI 83/84 calculators. Press 2 nd Vars, then #3 inverse Norm. Input the data in this way (Area to the left of the boundary, the mean, the standard deviation) Press ENTER.

42 The Central Limit Theorem (C.L.T.) 40 When we take repeated samples from a population and keep track of the sample means this is called a sampling distribution. Regardless of the shape of the original population the shape of the distribution of the sample means will become closer to a normal distribution as the sample size increases. The Central Limit Theorem (C.L.T.) states that when sampling from a large population of any distribution shape, the sample means will have a normal distribution whenever the sampling size is 30 or more. If samples are selected from a population that is normally distribution with a mean μ and standard deviation σ then the distribution of sample means will have a normal distribution and the mean of this distribution is μ = μ x and the standard deviation is σ x = σ n When do you use the Central Limit Theorem? You use the Central Limit Theorem whenever the sample size is greater than one. Sometimes the question will say Find the probability of the sample means or what is the probability that the mean length x. (Any unit can be used not just length). Important stuff: Check the flow chart on the next page to see if you can do the problem. μ x = μ Se σ x Standard error of the mean and σ x = σ n x-μ x z= σ n Once you know it is a CLT problem you need to decide if you can do the problem. The flow chart on the next page will help you make this decision.

43 41 Is the distribution normally distributed? YES NO Is the sample size 30 or more? YES NO You can do the problem You cannot do the problem The difference templates for the Normal distribution and the CLT Z x Z X µ x-μ Z= σ Normal Distribution x-μ Z= σ x x μ x CLT This is called the standard error of the mean σ = x σ n

44 Confidence Intervals for large numbers when sigma is known 42 A confidence interval is a range of values that is likely to contain the true value of the population parameter. For a given population μ and σ are called the parameters of the population. The degree of confidence is the probability (written as a decimal) that the confidence interval contains the true value of the population parameters. A point estimate is a single value or point used to approximate a population parameter. x is the point estimate for μ. Symbols used: C the symbol used for confidence interval (written as a decimal) E margin of error N sample size α (alpha) the significance level (the total area outside the confidence interval) How do I find the confidence interval given α? 100% - α (written as a %) α 2 Will give you the area in right or left tail which is what you need to find Z c In a confidence interval Z c is the critical region which is located at the right tail. Important stuff: N ; x ; σ ; C Calculations needed Z c σ E= n Round E up to the data What does a confidence interval look like??? Put the answer in this format: P({x E} < μ < { x+ E}) = C (written as a decimal) We are C confident that μ is in this region Given the critical value looking for E or x x = {upper CI limit + Lower CI limit} 2 E = {upper CI limit - Lower CI limit} 2

45 43 Confidence Intervals for small numbers when sigma is not known The student t distribution is used in performing inference for a population mean when the population being sampled is approximately normal and the population standard deviation is unknown. Degrees of freedom: The number of degrees for a data set corresponds to the number of scores that can vary after certain restrictions have been imposed on all the scores. Since the calculation of s depends on the value of x we have one restriction on the data. Therefore the number of degrees of freedom for this confidence interval will be n 1 Symbols used: d.f. degree of freedom n 1 (the number of variables free to change) C The symbol used for confidence interval (written as a decimal) E Margin of error N sample size t c is like the Z value at critical but it is called a t value (Use a t axis not a z axis) Flow chart for confidence intervals: C t c (Student t distributions with d.f. = n 1) Important stuff: N ; x ; s ; C Calculations needed t c s E = Round E to the data n Put the answer in this format: P({ x E} < μ < { x+ E}) = C (written as a decimal)

46 44 Use the flow chart on page 41 to determine if you will use Z c or t c then follow the steps below. Confidence Intervals large numbers or Z c Step 1 Find Z c using the conversion chart on page 46 Step 2 Find N, x and σ which is found in the problem Step 3 Calculate E using the formula Step 4 Arithmetic x ± E Z c σ E= n (Round to the data) Step 5 Write the C.I. in this format P({x - E} < µ < {x + E}) = C Confidence Intervals small numbers or t c Step 1 Find t c using the student t chart. Step 2 Find N, x and s which is found in the problem Step 3 Calculate E using the formula t c s E= n (Round to the data) Step 4 Arithmetic x ± E Step 5 Write the C.I. in this format P({x - E} < µ < {x + E}) = C

47 45 Using your TI 83/84 calculators to write a confidence interval large or small numbers: Large numbers: You can input the data manually ( x, σ, n and C) or place the data in L 1. a) Press Stat Tests Z interval. b) Use Data if you are putting the data in L 1 otherwise use Stats. c) Cursor down and fill in the information; (σ, x, n and C as a decimal) if you are using Stats, or List; Freq, C if you are using L 1. d) Cursor down to Calculate and press ENTER. The confidence interval will be in this format: ( x E, x+ E). Small numbers: You can input the data manually ( x, s, n and C) or place the data in L 1. a) Press Stat Tests t interval. b) Use Data if you are putting the data in L 1 otherwise use Stats. c) Cursor down and fill in the information; (s, x, n and C as a decimal) if you are using Stats, or List; Freq, C if you are using L 1. d) Cursor down to Calculate and press ENTER. The confidence interval will be in this format: ( x E, x+ E).

48 46 Confidence interval to Z c chart. You can use this conversion chart to find Zc given a confidence interval. CI Z c 70% % % % % % % % % % % How do I find Z c if it s not on the chart????? Take the confidence interval and subtract it from 100%, this is α the area in both tails, now divide this by 2 and you have the area in the left tail. Look this up in your negative Z table and you found Z c

49 47 Hypotheses testing when σ is known. (Large numbers) 1) Write the claim: 2) Write the claim and counter claim in symbolic form: Using μ and the symbol {= or or or or < or >} 3) Now label the claims H o or H a The claim containing {= or or } is labeled H o and stands for the Null hypotheses. The claim containing { or > or <} is labeled H a and stands for the Alternative hypotheses. 4) Make the normal curve and shade the tail. Remember the Alternative hypothesis identifies the tail { is two tails, > is a right tail and < is a left tail} 5) Now find Z c The significance level αis important, because if you have a right or left tail you need to double the α in order to get the C.I. and if you have a two tails you leaveαalone to get the C.I. 6) Use the formula x-μ Z= σ n to find the test statistic. 7) Go back to the normal curve and check to see if the test statistic falls in the critical region. 8) Follow the flow chart Does the original claim contain the equal sign? Yes Becomes H o Does the T.S. fall Yes Reject H o in the critical area? No Accept H o No Becomes H a Does the T.S. fall Yes Accept H a in the critical area? No Reject H a 9) Write you conclusion

50 Hypothesis testing large numbers σ is known 48 1) Write the claim in words. The claim can be found in the problem look for the words test the claim Claim: 2) Now write the claim and the counter claim in symbolic form. The counter claim is the opposite of the actual claim The claim and counter claim will use µ and one of these = ; ; ; ; < ; > µ µ 3) Now label both claims The claim containing {= or or } is labeled H o The claim containing { or < or >} is labeled H a H o H a 4) Make the Normal Distribution Curve and locate the critical region (the part that is shaded). Remember the Alternative hypothesis H a identifies the tail. is two tails, > is a right tail, and < is a left tail Z 0

51 5) Find the Z score at the critical region. Use the conversion chart on page 26 of the blue book. The significance levelαis important because if you have a right or left tail you need to double the α in order to get the C.I. and if you have two tails you leaveαalone to get the C.I. 6) Now compute the value of the test statistic. x-μ Z= σ n 49 7) Go back to the normal curve and check to see if the test statistic falls in the critical region. 8) Use the flow chart shown below. Does the original claim contain the equal sign? Yes Becomes H o Does the T.S. fall Yes Reject H o in the critical area? No Accept H o No Becomes H a Does the T.S. fall Yes Accept H a in the critical area? No Reject H a 9) State your conclusion using a complete paragraph.

52 The phone call How to get Z c in a hypothesis test large numbers 50 Hi Robbie, do you how to get Z c in a hypothesis test large numbers? Is the confidence interval given in the problem? Hi Marissa, yes I do. All you have to do is get the confidence interval then go to Page 46 of the blue book and use the conversion chart. Well yes and no. They give you α, the significance level, and all you have to do is subtract it from 100% and you have the confidence interval. That sounds easy but what is this doubling the alpha about If you have a left or right tail test you need to double the alpha. If you have a two tail test you leave alpha alone. How do I know if it s a right, left or two tailed test? That s simple too, just look at the alternate hypothesis (H a ) the inequality symbol will tell you what test it is. The > symbol is a right tailed test, the < symbol is a left tailed test and symbol is a two tailed test. What s this alternate hypothesis (H a ) you are talking about? The alternate hypothesis (H a ) is the claim that contains the symbol >, < or Thanks Robbie Glad I could help

53 51 The E mail on Hypothesis testing σ is known Hi Robbie, I hope you understand hypothesis testing I am so lost. This is like riding a bike the first time it seems hard but after a couple of tries it is easy. I hope so. Where do I start Robbie? Read the question and somewhere in the problem usually at the end it will ask you to test a claim. After you find this claim write it out in words in the box provided in step I of the template shown on page 48 of the blue book. This I can do, but how do I get this claim into symbolic form and what is the counter claim? When you write the claim it will contained one of these words. {Greater} ; {Less than} ; {Equal to} ; {Not equal to} ; {Greater than or equal to} or {Less than or equal to} rewrite the claim in box 2 of the template using µ, the symbol used to describe the words and the number given in the claim. The symbols you want are > ; < ; = ; ; ; The counter claim uses the opposite symbol of the one contained in claim. The opposite of > is ; the opposite of < is and the opposite of = is Robbie are you telling me if the claim said the mean temperature is greater than 80 F I write µ > 80 F? Then I bet the counter claim would be µ 80 F Good so far what s next? That s right Marissa. Good job Right again but I should point out most books and publications only use = but you can use in lieu of = until you get the hang of it. Now you need to label both claims H o and H a. This is easy because the claim that contains the equal sign is labeled H o which standards for the Null Hypothesis and the claim that does not have the equal sign is labeled H a which stands for the Alternate Hypothesis. This goes in box 3 of the template.

54 52 I know I need to make the normal curve now but how do I know what tail to shade? The Alternative hypothesis identifies the tail { is two tails, > is a right tail and < is a left tail} I understand now and you explained how to find Z c yesterday. So I put Z c at the critical region(s) and I will make Z c positive if it located at the right tail and negative if it is located at the left tail. Very good Marissa and don t forget if you have two tails use ± Z c I think I know what comes next. You calculate the value of the test statistic using the formula x-μ Z= σ Very good Marissa n Now that I have the value of the test statistic what do I do with it? Go back to the normal curve and check to see if the test statistic falls in the critical region Now I use the flow chart to find out which claim I accept or reject. Very good Marissa The last step is creative writing. That s right. Now that you know what claim to accept all you have to do is write a short sentence about that claim and you re done. Robbie do you think I should use the template on every question Yes I do. If you start cutting steps now you might make a mistake and this could cost you points on the next test OK see you in class tomorrow Bye Bye Marissa

55 53 Hi. I made the flow chart shown below to help you answer this question: Do I use a Z score or t score? Whenever you have Confidence Intervals or a hypothesis test question, you should follow this flow chart to decide if you should use a Z or t score. Start Here Yes. Is σ known? No. Is the data normally distributed? Is the data normally distributed? Yes. No. Yes. No. Yes. Is N 30? Yes. Is N 30? No. Use a Z score. Use a t score. You can t do the problem. No.

56 Hypothesis testing when σ is not known. (Small numbers) Remember: If the distribution is not normal and n < 30 you cannot do the problem 54 1) Write the claim: 2) Write the claim and counter claim in symbolic form: Uses μ and the symbol {= or or or or < or >} 3) Now label the claims H o or H a The claim containing {= or or } is labeled H o and stands for the Null hypotheses. The claim containing { or > or <} is labeled H a and stands for the Alternative hypotheses. 4) Make the normal curve and shade the tail. Remember the Alternative hypothesis identifies the tail { is two tails, > is a right tail and < is a left tail} 5) Now find t c Use the student t distribution table, the level of significance α and the degree of freedom to determine the t score at the critical region. The significance level αis important, because if you have a right or left tail you need to double the α in order to get the C.I. and if you have a two tails you leave α alone to get the C.I. x-μ 6) Use the formula t c = to find the test statistic. s n 7) Go back to the normal curve and check to see if the test statistic falls in the critical region. 8) Follow the flow chart Does the original claim contain the equal sign? Yes Becomes H o Does the T.S. fall Yes Reject H o in the critical area? No Accept H o No Becomes H a Does the T.S. fall Yes Accept H a in the critical area? No Reject H a 9) Write you conclusion

57 Hypothesis testing small numbers σ is NOT known 55 1) Write the claim in words. The claim can be found in the problem look for the words test the claim Claim: 2) Now write the claim and the counter claim in symbolic form. The counter claim is the opposite of the actual claim The claim and counter claim will use µ and one of these = ; ; ; ; < ; > µ µ 3) Now label both claims The claim containing {= or or } is labeled H o The claim containing { or < or >} is labeled H a H o H a 4) Make the Normal Distribution Curve and locate the critical region (the part that is shaded). Remember the Alternative hypothesis H a identifies the tail. is two tails, > is a right tail, and < is a left tail t 0

58 5) Find the t score at the critical region. Use the T distribution chart in the back of the blue book. The significance levelαis important because if you have a right or left tail you need to double the α in order to get the C.I. and if you have two tails you leaveαalone to get the C.I. 6) Now compute the value of the test statistic 56 t = c x-μ s n t c = 7) Go back to the normal curve and check to see if the test statistic falls in the critical region. 8) Use the flow chart shown below. Does the original claim contain the equal sign? Yes Becomes H o Does the T.S. fall Yes Reject H o in the critical area? No Accept H o No Becomes H a Does the T.S. fall Yes Accept H a in the critical area? No Reject H a 9) State your conclusion using a complete paragraph.

59 57 Correlation Coefficient In statistics the correlation coefficient indicates the strength and direction of a linear relationship between two random variables. r The linear correlation coefficient for a sample Formula r= n xy - x y 2 2 n y2 - y n x - x * 2 Notation n The number of data pairs x The sum of the x values y The sum of the y values x 2 The sum of x squared ( x) 2 The x values should be added up then the sum is squared x y Multiply x and y then add the products y 2 The sum of y squared ( y) 2 The y values should be added up then the sum is squared

60 Positive correlation: If x and y have a strong positive linear correlation, r is close to +1. An r value of exactly +1 indicates a perfect positive fit and this occurs only when the data points all lie exactly on a straight line. A correlation greater than.8 is described as strong and a correlation less than.5 is described as weak. 58 Correlation r is close to 1 Y X Negative correlation: If x and y have a strong negative linear correlation, r is close to - 1. An r value of exactly -1 indicates a perfect negative fit and this occurs only when the data points all lie exactly on a straight line. Correlation r is close to -1 Y X

61 Zero correlation: If there is no linear correlation or a weak linear correlation, r is close to 0. A value near zero means that there is a random, nonlinear relationship between the two variables 59 Correlation r is close to 0 Y correlation, r is close to 0 correlation, X

62 60 Linear regression equation made easy (Sample) y = mx + b x is the independent variable m is the slope and b is the y intercept. Make a table like this X Y x y x 2 The x coordinates go here. The y coordinates go here. Multiply the x and y coordinates. Square the x coordinate. x y x y x 2 Formula to find the slope n xy - x y m= n x2-2 x Formula to find the y intercept the easy way b = y-mx To findy find the mean of the y coordinates. To find x find the mean of the x coordinates. OR you can use this formula y x2 - x xy b= n x2-2 x

63 61 Hi. Let me show you how to get all the information you need to find m and b in a linear regression equation. a) Put the x coordinates in L 1 and the y coordinates in L 2 b) Press Vars. c) Press # 5 Statistics, cursor over to. d) Select x and press ENTER. e) Select the other options one at a time. f) Substitute this information into the formulas, do the arithmetic and you found m and b. If you want the linear regression equation follow these steps: a) Press Stat. b) Cursor over to Calc. Press #4 LinReg (ax + b). c) Press ENTER.