A.REPRESENTATION OF DATA

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1 A.REPRESENTATION OF DATA (a) GRAPHS : PART I Q: Why do we need a graph paper? Ans: You need graph paper to draw: (i) Histogram (ii) Cumulative Frequency Curve (iii) Frequency Polygon (iv) Box-and-Whisker Plot (b) FREQUENCY DISTRIBUTION TABLE Q: How do I know it is a frequency distribution table? Ans: It says so on the QUESTION! It looks like a table with frequency on it. HENCE, FREQ distribution TABLE Q: What is it used for? Ans: You use the frequency distribution table to calculate: *no need to change interval. Applied for any data (i) Mean (ii) Standard Deviation It can also be used to calculate the following: *must change interval if data is continuous! (i.e. nearest to..) (iii) Frequency Density (iv)cumulative Frequency so that you are able to draw the CORRECT HISTOGRAM AND CUMULATIVE FREQUENCY CURVE Q: What is the formula of mean and standard deviation used on a freq distribution table? fx fx 2 mean = standard deviation = x 2 f f Q: How do I calculate the mean and standard deviation from a freq. dist table that looks like: () Frequency Distribution Table that has NO INTERVAL. x Frequency Directly multiplied the x values with the corresponding frequency.

2 mean = (2 4) + (3 6) + (5 5) + (8 ) = OR for careful calculation, you use the following table: x x 2 f fx fx SUM: therefore, standard deviation = (2) Frequency Distribution Table that has INTERVAL. x Frequency We need to calculate the exact value of x i.e. mid-value so that we can use the formula of mean and standard deviation. x (interval) x (mid-value) x 2 f fx fx SUM: mean = standard deviation = 2

3 (c) GRAPHS : PART II Q: What do I need to do BEFORE drawing the graphs listed in part I? Ans: IDENTIFY YOUR DATA Check whether your data is discrete or continuous Q: How do I know if my data is discrete or continuous? Ans: You know just by reading the question. If there s a word correct to the nearest..., data is continuous. Otherwise, it is a discrete data. Q: If my data is discrete, what s next? Ans: CHECK YOUR INTERVAL () Each interval has a gap in between (discrete) x Frequency For ALL GRAPHS, you must fix the interval so that there is NO gap. x Frequency (i) Histogram : Only AFTER YOU FIX NO GAP, you can calculate the frequency density. x Frequency Class Width Frequency Density 3

4 (ii) Cumulative Frequency Curve : Only AFTER YOU FIX NO GAP, you need to fix the interval by changing it into < upper boundary of your NEW interval and only then you can calculate the cumulative frequency x Frequency x < 2 < 5 < 8 < < 6 Cumulative Frequency (2) If there is no gap, you don t need to fix anything. Q: If my data is continuous, what s next? Ans: CHECK YOUR INTERVAL () Each interval has a gap in between (continuous) correct to the nearest... x Frequency For ALL GRAPHS, you must fix the interval so that there is NO gap. x Frequency (i) Histogram : Only AFTER YOU FIX NO GAP, you can calculate the frequency density. x Frequency Class Width Frequency Density 4

5 (ii) Cumulative Frequency Curve : Only AFTER YOU FIX NO GAP, you need to fix the interval by changing it into < upper boundary of your NEW interval and only then you can calculate the cumulative frequency x Frequency x <.5 < 4.5 < 7.5 < 0.5 < 5.5 Cumulative Frequency (2) If there is no gap, you don t need to fix anything. Q: After I fix the interval and calculate the necessary values, how do I make sure that the graph is correct? Ans:. Correct width for each interval in the x-axis as well as the y-axis 2. Labelled your axis according to the question 3. Use crosses for each value. DO NOT USE DOTS! 4. Make sure your CUMULATIVE FREQUENCY CURVE is a S-SHAPED For example: Correct x-axis Wrong x-axis Correct y-axis Wrong y-axis 5

6 (b) CODING Q: When do I use the coding method? Ans: When it is in the form such as below: (x 3) = 25 (x 3) 2 = 85 Q: What am I calculating? Ans: You want to calculate the mean of x and the standard deviation of x. Q: So what do I do first? Ans: Get all the equations you need by SUBSTITUTING x 3 = y Then you get the following: CODING TO USE mean: x 3 = ȳ (i) (just add the bar ) standard deviation: SD x = SD y (ii) (always equal) Q: How do I calculate the mean of x and standard deviation of x? Ans: You must calculate the mean of y and standard deviation of y first using the formula for mean and standard deviation from your formula booklet. Then by substituting these values into equation (i) and (ii) above, you get your answers. Working: 6

7 (c) STEM AND LEAF (STEM PLOT) Q: What do I do? Ans: List down the following details (i) Smallest value in your data (ii) Largest value in your data Next, you think about the appropriate interval which has the same class width based on the smallest and largest value. 25!! 4!! 8!! 23!! 30!! 2 6!! 20!!!! 4!! 35!! 27 8!! 3!! 2!! 7!! 2!! Smallest value = Largest value = Stem will represent your tens Leaf will represent your units Interval Stem Leaf Next you have to rearrange the values in ascending order and most importantly, you must show your key Stem Leaf Interval 7

8 (d) MEDIAN AND INTERQUARTILE RANGE Q: How do I calculate median and interquartile range from the following: (i) Stem and leaf If your number of data n = odd, you can use the formula below to find the position: median = lower quartile = upper quartile = 2 (n + ) 4 (n + ) 3 (n + ) 4 After you get the position, you count in ascending order from the top. Working: (ii) Cumulative Frequency Curve You get the value of your median and interquartile from the curve. Just calculate: median = total data or frequency from the x-axis 2 lower quartile = total data or frequency from the x-axis 4 3 upper quartile = total data or frequency from the x-axis 4 You go from the value you just calculated on the x-axis...hit the curve...then go to the y-axis. Working: ON THE GRAPH PAPER! (e) BOX-AND-WHISKER PLOT Q: What do I need? Ans: All you need is the following: (i) Smallest value of your data (ii) Largest value of your data (iii) Median (iv) Lower quartile (v) Upper quartile Working: ON THE GRAPH PAPER! 8

9 B. PROBABILITY (a) BASIC PROBABILITY FORMULA Q: How do I study this? Ans: All you need to do is memorize and UNDERSTAND what each of the following formula represents. Let A and B be events. P (A B) = If A and B are independent, P (A B) = If A and B are mutually exclusive, P (A B) = Conditional Probability, P (A B) = P (A B P (B) Q: How do I know events A and B are independent? Ans: By understanding the events given in the question. A and B are using different objects not connected in any way. A real-life example is the following. Consider a fair coin and a fair six-sided die. Let event A be obtaining heads, and event B be rolling a 6. Then we can reasonably assume that events A and B are independent, because the outcome of one does not affect the outcome of the other. Q: How do I know events A and B are mutually exclusive? Ans: By understanding the events given in the question. A and B are same object but different outcomes. An example of a mutually exclusive event is the following: Consider a fair six-sided die as before, only in addition to the numbers through 6 on each face, we have the property that the even-numbered faces are colored red, and the odd-numbered faces are colored green. Let event A be rolling a green face, and event B be rolling a 6. But it is obvious that events A and B cannot simultaneously occur. This makes sense because if A and B are mutually exclusive, then if A occurs, then B cannot also occur; and vice versa. 9

10 (b) PROBABILITY TREE Q: How do I know when to apply a probability tree? Ans: Read and UNDERSTAND the question! HAH! OR there are hints. For example, there are TOO MANY (or at least three) PROBABILITY VALUES INFORMATION in your question. Q: How do I create the probability tree? Ans: REMEMBER! THERE IS ONLY ONE TREE and MANY BRANCHES! Suppose you have 3 boxes noted as A, B and C. Box A contains 3 red and 4 green balls while box B contains 2 red and 6 green. Meanwhile box C contains only 3 green balls. A box is chosen at random and two balls are drawn with no replacement. Draw the probability tree. Working: REMEMBER! NO REPLACEMENT means you have to MODIFY the PROBABILITY IN THE PROBABILITY TREE! 0

11 Q: How do I calculate the probability from a probability tree? Ans: Always calculate from the beginning until the end of the branch. If you are travelling using a continuous route, you multiplied the values. If you are travelling from one option to the next option, you add after you multiplied the continuous route. Working: A box is chosen at random. Find the probability that both balls are green. Q: How do I know if the question is a conditional probability? Ans: Normally, it ll mention the word conditional probability or given that... One way of disguising the question is for example: The two balls drawn are red. Find the probability that they both came from box A Q: Which is which? Box A on the left side...or...two balls are green on the left hand side...vice versa. Ans: The two balls drawn are red. Find the probability that they both came from box A P (A B)

12 Given that the balls drawn are from box A, find the probability that both balls are red P (A B) Find the conditional probability for box A given that both balls are red P (A B) 2

13 LICENSE TO DRAW Below are following questions where you have to draw the graph such as HISTOGRAM, CUMULATIVE FREQUENCY CURVE, FREQUENCY POLYGON AND BOX PLOT on graph paper. () Decide which graph it is. (2) Draw up the APPROPRIATE TABLE such as FREQUENCY DISTRIBUTION TABLE, CUMULATIVE FREQUENCY TABLE (3) Show the APPROPRIATE CALCULATION such as FREQUENCY DENSITY, CUMULATIVE FREQUENCY, MEDIAN, INTERQUARTILES etc. Answer the following questions:. Look at the exercise given to you in class before the Holidays. (Paper 6 Revision). Which question requires you to use a graph paper? 2. In a competition to grow the tallest hollyhock, the heights recorded by 50 primary school children were as follows. Heights were measured to the nearest centimetre. Height (cm) Frequency Calculate the mean and standard deviation. Draw a histogram and the frequency polygon in one graph. 3. The times, to the nearest minute, taken by a group of 20 students to write a particular essay, were recorded and are grouped in the table below. Time (minutes) Number of Students Calculate the mean and standard deviation. Draw a cumulative frequency curve. Use your curve to estimate (i) the interquartile range of the times, (ii)the percentage of these students who spent over 62 minutes in writing the essay. 4. The patients at a chest clinic were asked to keep a record of the number of cigarettes they smoked each day. Number of cigarettes smoked per day and over Frequency Draw a histogram to represent this data. 5. The table below gives the lengths, in minutes, of 50 telephone calls from a school office. Length of call (mins) Number of calls (a) Draw a cumulative frequency curve. (b) Estimate the median and the quartiles (c) Draw a box plot.

14 Extra Class st Term EXERCISE ON REPRESENTATION OF DATA AND PROBABILITY. The average height of 20 boys is 60 cm, with a standard deviation of 4 cm. The average height of 30 girls is 55 cm, with a standard deviation of 3.5 cm. Find the standard deviation of the whole group of 50 children. 2. A teacher recorded the times taken by 2 boys to swim one length of the pool. The times are given to the nearest second Draw a stem and leaf diagram to illustrate the results. Find the median and the interquartile range. 3. For a particular set of data, n = 00 (x 50) = 23.5 (x 50) 2 = Find the mean and standard deviation of x. 4. Find the mean length for the data represented by the stem and leaf diagram. Stem Leaf Two ordinary fair dice, one red and one blue, are to be rolled once. (a) Find the probability of the following events: Event A : the number showing on the red die will be a 5 or a 6 Event B: the total of the numbers showing on the two dice will be 7 Event C: the total of the numbers showing on the two dice will be 8 (b) State, with a reason, which two of the events A, B and C are mutually exclusive. (c) Show that the events A and B are independent. 6. I travel to work by route A or route B. The probability that I choose route A is The probability that I am late for 4. 2 work if I go via route A is and the corresponding probability if I go via route B is 3 3. (a) What is the probability that I am late for work on Monday? (b) Given that I am late for work, what is the probability that I went via route B?

15 Extra Class st Term 7. A box contains 20 chocolates, of which 5 have soft centres and five have hard centres. Two chocolates are taken at random, one after the other. Calculate the probability that (a) both chocolates have soft centres (b) one of each sort of chocolate is taken (c) both chocolates have hard centres, given that the second chocolate has a hard centre. 8. The probabilities of events A and B are P(A) and P(B) respectively. P (A) = 5 2, P(A B) =, P(A B) =q 6 Find, in terms of q, (a) P(B) (b) P(A B) Given that A and B are independent events, (c) find the value of q. 9. A school has three photocopiers A, B and C. On any given day the independent probabilities of a breakdown are 0. for A, 0.05 for B and 0.04 for C. For a randomly chosen day, calculate the probability that (a) at least one of the copiers breaks down (b) exactly one of the copiers breaks down (c) given exactly one copier breaks down, then it is copier C.

16 C. DISCRETE PROBABILITY DISTRIBUTION Q: What is DISCRETE? Ans: It is just EXACT DATA. EXACT VALUES. NO ESTIMATIONS OR APPROXIMATIONS. For example, your shoe size, your age. (a) PROBABILITY DISTRIBUTION TABLE Q: How does the table look like? Ans: It has PROBABILITY values (usually in fractions or decimal) on the table. For example, x P ( X = x ) Q: Where does the table come from and how do I know it s correct? Ans: Usually it comes from the probability tree. Or simply from the question itself by listing down the probability for each outcome. For example, a die is thrown once. Let X be the random variable for the number shown on the die. Find the probability distribution for X. Q: How do I create the table and how do I know it s correct? Working: 3

17 Q: How do I answer the next question? x P ( X = x ) a Find a. Q: What can I calculate from the probability distribution table? Ans: The expectation E(X) or µ and the variance Var(X) (as well as the standard deviation) Q: How do I calculate mean and the variance using the table? Ans: Use the formula below: E(X) = xp (X = x) Var (X) = x 2 P (X = x) (mean) 2 = E(X 2 ) [E(X)] 2 = E(X 2 ) µ 2 Standard deviation = Var(X) x P ( X = x ) Working: 4

18 D. BINOMIAL DISTRIBUTION (a) CHARACTERISTICS Q: What is the property of Binomial Distribution? Ans: (i) There are only two outcomes; success (p) and failure ( -p = q ) (ii) The trials are finite and independent (b) FORMULA Q: What is the formula for calculating the probability in binomial question? Ans: On the formula booklet that states for the binomial distribution B(n,p). You can use the formula below as well: P (X = r) = n C r p r q n r Q: When do I use this formula? Ans: You use this formula when you identify the question is BINOMIAL! DUH!...or you can detect from the question, there s a word exactly. Q: How do I calculate the probability using the formula? Ans: List down the values of n, p, q and X (what event is it?). Suppose we have the following: Values B (n, p) Calculate P (X???) n = 6, p = 0.4, q = n = 3, q = 0.3, p = n=4, p = 0.2, q = 0.8 Calculate the probability of at exactly 3 apples. Calculate the probability of more than two ciku. Calculate the probability of at least one durian. Working: 5

19 (c) EXPECTATION AND STANDARD DEVIATION Q: What is the formula to calculate the expectation and standard deviation for binomial? Ans: Unlike DISCRETE PROBABILITY DISTRIBUTION, we have to use the probability distribution table to calculate the expectation and the standard deviation. For BINOMIAL, we just use the EASIEST FORMULA: Expectation E(X) = np Variance = npq Standard Deviation = (c) DETECTION Q: How do I know if the question is BINOMIAL? Ans: It is very easy to detect if the question is binomial. There is only two probabilities available in the question. A failure and a success of the event. The easiest detection is look at the number of trials. If it is not BIG, it is binomial. For example, Kamal has 30 hens. The probability that any hen lays an egg on any day is 0.7. Hens do not lay more than one egg per day, and the days on which a hen lays an egg are independent. Calculate the probability that, on any particular day, Kamal s hens lay exactly 24 eggs. 6

20 E. NORMAL DISTRIBUTION (a) DETECTION Q: How do I know it is a normal question? Ans: Usually, there s a word normal in the question. or normally distributed. But the best hint they ll give is the number of trials you are using. The n is VERY BIG! (b) PROBABILITY IN NORMAL Q: What do I need to do calculate the probability in normal? Ans: Follow the THREE steps below: () TRANSLATE YOUR QUESTION INTO P (X?????) This is very crucial. If you misunderstood the question, the whole working is wrong. (2) CHANGE INTO Z The formula change into Z : this one! Z = x µ σ x the value on your translation P(X... x) µ the mean σ standard deviation. REMEMBER! If variance is given, you have to calculate standard deviation. Question P (X???) Change into Z A farm which produced apples is normally distributed with mean 0 and the standard deviation 20. Find the probability when the apples are less than 50. A farm normally produced durians with mean 50 and variance 00. Find the probability when the durians produced is more than 30. (3) CHANGE P(Z...) INTO P(Z <...) if needed Q: Why do I need to change into the correct inequality sign? Ans: This is because we can only read the NORMAL TABLE if the probability has < sign. φ(a) P (Z <a)=φ(a) P (Z >a)= P (Z <a) = φ(a) - the probability a. Find the probability of a from the table. 7

21 Q: How do I change the sign if my P (Z...) are as follows? P(Z...) P(Z <...) P (Z >.34) P (Z >.34) P (Z <.34) P (.34 <Z<2) P (.34 <Z<2) P ( 2 <Z<.34) (c) NOT A PROBABILITY QUESTION Q: What if the question is not asking for probability? Ans: What do you mean? Q: For example, P (Z <z)=0.8533, find z. Ans: You will have to use the reverse method. Again you will have to go through step () on the previous page. i.e. translate the question. Question P (Z <z)= P (Z >z)= P (Z <z)=0.23 P (Z >z)=0.635 Find the smallest value of n if there is a probability of at least 0.85 that a random sample of n tapes contains at least one damaged tape. 8

22 (d) APPROXIMATION Q: How do I know when to use approximation method? Ans: When the question says... using an appropriate or suitable approximation or the number of trials is very very BIG! You can check this by applying the condition below: np > 5 nq > 5 both YES! NOT only one of them Q: How do I use approximation method? Ans: Just follow the steps below: () LISTING DOWN EVERYTHING YOU HAVE AND CALCULATE MEAN AND STANDARD DEVIATION Values CALCULATE µ = np σ = npq n p q (2) INTERPRET YOUR QUESTION INTO P(X...) (3) APPLY CONTINUITY CORRECTION C.C Q: How do I apply continuity correction? Ans: The easiest way is to think about a real number line. What s included...what s not. BEFORE PROCESS AFTER P (X 30) P (X <28) P (X 38) P (X >55)

23 BEFORE PROCESS AFTER P (8 <X<29) P (3 X 63) P (0 <X 9) (4) CHANGE INTO Z (5) CHANGE P(Z...) INTO P(Z <...) if needed (e) FINDING µ OR σ OR BOTH Q: How do I calculate the mean and standard deviation in a normal question? Ans: Usually, the question gives at least one probability already. For example, the random variable of X is distributed normally with N(µ, 2). It is known that P (X >32) = and the variance is 2. Find the value of µ. Working: 20

24 WHICH IS WHICH?! Extra Class st Term Here, we are going to look at the questions, UNDERSTAND THE QUESTION, INTERPRET THE QUESTION and most importantly IDENTIFY WHAT TYPE OF DISTRIBUTION IT IS. Draw the table below and FILL in each of the columns after you READ the QUESTION below the table. Q NO. KEYWORDS DISTRIBUTION D B N INTERPRET INTO P(X...) if B or N NEXT STEP? If your question have parts, interpret each part as follows... (a) (b) etc. Answer the following question:. A discrete random variable X has the following probability distribution and can only take the values tabulated. X 3 6 n 2 Probability k (a) Find the value of k Given that E(X)=6.0, find (b) the value of n (c) the variance of X. 2. The masses of boxes of apples are normally distributed such that 20% of them are greater than 5.08 kg and 5% are greater than 5.62 kg. Estimate the mean and standard deviation of the masses. 3. A certain tribe is distinguished by the fact that 45% of the males have six toes on their right foot. Find the probability that, in a group of 200 males from the tribe, more than 97 have six toes on their right foot. 4. A bag contains five black counters and six red counters. Two counters are drawn, one at a time, and not replaced. Let X be the number of red counters drawn. Find E(X). 5. A bag contains counters of which 40% are red and the rest yellow. A counter is taken from the bag, its colour noted and then replaced. This is performed eight times in all. Calculate the probability that (a) exactly three will be red (b) at least one will be red (c) more than four will be yellow.

25 WHICH IS WHICH?! Extra Class st Term 6. Cartons of mil from a particular supermarket are advertised as containing litre, but in fact the volume of contents is normally distributed with a mean of 02 ml and a standard deviation of 5 ml. (a) Find the probability that a randomly chosen carton contains more than 00 ml. (b) In a batch of 000 cartons, estimate the number of cartons that contain less than the advertised volume of milk 7. 0% of the articles from a certain production line are defective. A sample of 25 articles is taken. Find the expected number of defective items and the standard deviation. 8. The lengths of metal strips are normally distributed with a mean of 20 cm and a standard deviation of 0 cm. Find the probability that a strip selected at random has a length (a) greater than 05 cm (b) within 5 cm of the mean Strips that are shorter than L cm are rejected. Estimate the value of L, correct to one decimal place, if 9% of all strips are rejected. 9. The random variable X is distributed B(200, 0.7). Use the normal approximation to the binomial distribution to find (a) probability of X is more than or equal to 30 (b) P (36 X<48) (c) P (X <42) (d) probability exactly 52

Statistics (This summary is for chapters 17, 28, 29 and section G of chapter 19)

Statistics (This summary is for chapters 17, 28, 29 and section G of chapter 19) Mean, Median, Mode Mode: most common value Median: middle value (when the values are in order) Mean = total how many = x