Normal Probability Distributions

Transcription

1 C H A P T E R Normal Probability Distributions 5 Section 5.2 Example 3 (pg. 248) Normal Probabilities Assume triglyceride levels of the population of the United States are normally distributed with a mean of 134 and a standard deviation of 35. If you randomly select one American, calculate the probability that his/her triglyceride level is less than 80, that is P(X < 80). The calculator function normalcdf(lowerbound, upperbound, μ, ) computes probability between a lowerbound and an upperbound. In this example, you are computing the probability to the left of 80, so 80 is the upperbound. In examples like this, where there is no lowerbound, you can always use negative infinity as the lowerbound. One way to indicate negative infinity is -1 x 10^99. On the TI-84, use (-) 1 2 nd [EE] 9 9 (Note: EE is the notation used on TI-84 for raising 10 to a power. It is the second function on the comma key:, ). Try entering -1 x 10^99 by pressing (-) 1 2 nd [EE] 9 9 into your calculator. The result, -1 x 10^99, is displayed as -1E99: Now, to calculate P(X < 80), press 2 nd [DISTR] and select 2:normalcdf( If you have the STAT WIZARD turned On, you will see the following menu: Notice that the lower bound of negative infinity is the default and is displayed as -1E99. Also notice that the values for and for a standard normal random variable are also displayed. Enter the value for the upper bound of the region whose area you are calculating. That value is 80. Enter the correct values for and. Scroll down to Paste and The values from the Stat Wizard menu are now pasted into the normalcdf function. 63

2 64 Chapter 5 Normal Probability Distribution If you have the STAT WIZARD turned Off, once you press 2 nd [DISTR] and select 2:normalcdf( type in -1E99, 80, 134, 35 ) and Technical note: Theoretically, the normal probability distribution (the bell-shaped curve) extends infinitely to the right (positive infinity) and to the left (negative infinity) of the mean (see Textbook pg. 234). In this particular problem, P(X < 80), you do not necessarily have to use negative infinity (-1 E 99) as your lowerbound. If you look at this example in your textbook on pg. 248, the lowerbound is set at This is a perfectly fine selection since falls far below any possible minimum triglyceride level for an individual. Notice that your results are the same in both of the above screens.

3 Section Exercise 7 (pg. 249) Heights of Males - Finding Probabilities In this exercise, use a normal distribution with μ = 69.4 and = 2.9. a. To find P(X < 66), press 2 nd [DISTR], select 2:normalcdf( If you have the STAT WIZARD turned On, paste the appropriate values into the Menu, scroll down to Paste and If you have the STAT WIZARD turned Off, once you select 2:normalcdf( enter -1E99, 66, 69.4, 2.9 ) and (Note: Since no male will have a height less than 0 inches, you could use 0 in place of 1 E 99 as your lowerbound. b. To find P(66< X < 72), press 2 nd [DISTR], select 2:normalcdf(, use 66 as the lowerbound and 72 as the upperbound. c. To find P(X > 72), press 2 nd [DISTR], select 2:normalcdf(, and use 72 as the lowerbound and 1E99 as the upperbound. (Note: In this example, the lowerbound is 72 and the upperbound is positive infinity). d. None of the outcomes are unusual since the probabilities are greater than 0.05 in each case. Note: When using the TI-84 (or any other technology tool), the answers you obtain may vary slightly from the answers that you would obtain using the standard normal table. The differences are simply due to rounding.

4 66 Chapter 5 Normal Probability Distribution Section 5.3 Example 4 (pg. 255) Finding a Specific Data Value This is called an inverse normal problem and the command invnorm( area, μ, ) is used. In this type of problem, a percentage of the area under the normal curve is given and you are asked to find the corresponding X-value. In this example, the percentage given is the top 10%. In the invnorm command, the area must be the area from negative infinity up to the specified X-value. To find the X-value corresponding to the top 10%, you must accumulate the bottom 90% of the area. The area value must be entered in decimal form. Press 2 nd [DISTR] and select 3:invNorm( If you have the STAT WIZARD turned On, you will see the following menu: Enter.90 for the area, enter the values for μ and, scroll down to Paste and The values from the Stat Wizard menu are now pasted into the invnorm function. If you have the STAT WIZARD turned Off, once you press 2 nd [DISTR] and select 3:invNorm( type in.90, 50, 10 ) and In order to score in the top 10%, you must earn a score of at least Assuming that scores are given as whole numbers, your score must be at least 63.

5 Section Exercise 32 (pg. 258) Heights of Males Finding A Specific Value This is a normal distribution with μ = 69.4 and = 2.9. a. To find the 90 th percentile, press 2 nd [DISTR] and select 3:invNorm( If you have the STAT WIZARD turned On, paste the appropriate values into the Menu, scroll down to Paste and If you have the STAT WIZARD turned Off, once you select 3:invNorm( type in.90, 69.4, 2.9 ) and b. To find the first quartile, press 2 nd [DISTR] and select 3:invNorm( and type in.25, 69.4, 2.9 ) and 67

6 68 Chapter 5 Normal Probability Distribution Section 5.4 Example 4 (pg. 266) Probabilities for Sampling Distributions In this example, data has been collected on the average daily driving time for different age groups. From the graph on pg. 266, you will find that the mean driving time for adults in the 15 to 19 age group is: = 25 minutes. The problem states that the assumed standard deviation is = 1.5 minutes. You randomly sample 50 drivers in the age group. Since the sample size, n, is greater than 30, you can conclude that the sampling distribution of the sample mean is approximately normal with = 25 and =. To calculate P( 24.7 < x < 25.5), press 2 nd [DISTR], select 2:normalcdf( If you have the STAT WIZARD turned On, paste the appropriate values into the Menu, scroll down to Paste and Note: The entry for must be the standard deviation of the sample means which is and is equal to. If you have the STAT WIZARD turned Off, once you select 2:normalcdf(, type in 24.7, 25.5, 25, ) and Note: The answer in your textbook is This answer was calculated using the z-table. Since z-values in the table are rounded to hundredths, the answers will vary slightly from those obtained using the TI

7 Section Example 6 (pg. 268) Finding Probabilities for x and x The population is normally distributed with μ= 3173 and = To calculate P(X < 2700), press 2 nd [DISTR], select 2:normalcdf(. If you have the STAT WIZARD turned On, paste the appropriate values into the Menu, scroll down to Paste and If you have the STAT WIZARD turned Off, once you select 2:normalcdf(, type in -1E99, 2700, 3173, 1120 ) and (Note: Since the minimum credit card balance is 0, the lowerbound could be set at 0, rather than negative infinity.) 2. To calculate P( x < 2700), press 2 nd [DISTR], select 2:normalcdf( and type in -1E99 for the lowerbound, 2700 for the upperbound, = 3173 and = ) and

8 70 Chapter 5 Normal Probability Distribution Exercise 35 (pg. 272) Paint Cans Finding Probabilities To decide whether the machine needs to be reset, you must decide how unlikely it would be to find a mean of from a sample of 40 cans if, in fact, the machine is actually operating correctly at μ = 128. One method of determining the likelihood of x = is to calculate how far is from the mean of 128. You can do this by calculating how much area there is under the normal curve to the left of The smaller that area is, the farther is from the mean and the more unlikely is. To calculate P( x 127.9), press 2 nd [DISTR], select 2:normalcdf( If you have the STAT WIZARD turned On, paste the appropriate values into the Menu, scroll down to Paste and If you have the STAT WIZARD turned Off, once you select 2:normalcdf(, type in --1E99, 127.9, ) and (Note: Since the minimum amount of paint is 0, the lowerbound could be set at 0, rather than negative infinity.) Notice that the answer is displayed in scientific notation: 7.827E-4. Convert this to standard notation, , by moving the decimal point 4 places to the left. This probability is extremely small; therefore, the event ( x 127.9) is highly unlikely if the mean is actually operating correctly with μ = 128. So, something has gone wrong with the machine and the actual mean must have shifted to a value less than 128.

9 Technology 71 Technology (pg. 293) Age Distribution in the U. S. 1: Press STAT and select 1:EDIT. Clear L1, L2 and L3. Enter the age distribution from the table into L1 and L2 by putting class midpoints in L1 and relative frequencies (converted to decimals) into L2. (The first entry is 2 in L1 and.069 in L2.) To find the population mean, μ, and the population standard deviation,, press STAT, highlight CALC, select 1:1- Var Stats. If you have the STAT WIZARD turned On, make sure that L1 is selected for the List. Enter 2 nd FreqList and scroll down to Calculate and. [L2] for. If you have the STAT WIZARD turned Off, when you highlight CALC and select 1:1-Var Stats and press ENTER,, you must enter the names of the 2 columns that contain the information. So, press 2 nd [L1], 2 nd [L2] and press ENTER to see the descriptive statistics. The mean and the standard deviation will be displayed. (Note: Use x because the age data represents the entire population distribution of ages, not a sample.) 2: Enter the thirty-six sample means into L3. To find the mean and standard deviation of these sample means, press STAT, highlight CALC, select 1:1-Var Stats. Press ENTER and press 2 nd [L3] ENTER. The mean and the standard deviation will be displayed. (Note: use sx because the 36 sample means are a sample of 36 means, not the entire population of all possible means of size n = 40). 3: Use the histogram on pg. 293 to answer this question. 4: The TI-84 will draw a frequency histogram for a set of data, not a relative frequency histogram. (The shape of the data can be determined from either type of histogram). Press 2 nd [STAT PLOT] and select 1: Plot 1. Turn ON Plot 1, select Histogram for Type, set Xlist to L3 and set Freq to 1. Press ZOOM. If you have the STAT WIZARD turned On, you may see the following screen:

10 72 Chapter 5 Normal Probability Distribution Press CLEAR to indicate that you do not wish to use any of the options displayed at the bottom of the screen. Press ZOOM again and you will see the Zoom screen. Select 9 for ZoomStat. If you have the STAT WIZARD turned Off, when you press ZOOM you will immediately see the Zoom screen. Select 9 for ZoomStat. To adjust the histogram so that it has nine classes, press WINDOW. For Xmin, use a value slightly smaller than the minimum value of the sample means; for Xmax, use a value slightly larger than the maximum value of the sample means; approximate the class width by finding the range of values (max min) of the sample means and dividing this range by 9. This value is approximately 2, so set Xscl = 2 and press GRAPH. 5: See the output from Exercise 1 for the population standard deviation,. 6: See the output from Exercise 2 for the standard deviation of the 40 sample means. This standard deviation,, is an approximation of. The Central Limit Theorem states that. Use your results to confirm this fact.