(b) Calculate the probabilities of the events A, B and C given in the previous problem.

Chapter 1 Lecture Examples 1. Find the area under the snc to the right of: z = 1.82; z = −2.11; z = −0.09.

9. You are given the following information: the events A and B are disjoint; P (A) = 0.40; and P (B) = 0.25. Calculate the following probabilities.

2. Find the area under the snc to the left of: z = 0.82; z = −1.19; z = −0.92.

(a) P (A or B). (b) P (Ac ). (c) P (B c ).

3. Find the area under the snc to the right of z = 1.4238.

10. You are given the following information: P (A) = 0.25; P (B) = 0.45; P (AB) = 0.20. Calculate P (A or B).

4. Find the area under the snc to the right of: z = 3.72; z = −3.89.

11. What is wrong with each of the following?

5. Find the area under the snc to the left of:

(a) P (A) = 0.20; P (B) = 0.55; and P (AB) = 0.25.

z = −3.90; z = 4.19.

(b) P (A) = 0.60; P (B) = 0.55; and A and B are disjoint.

6. It is possible to purchase four-, eight-, ten-, twelve-, and twenty-sided dice on the internet, in addition to the usual six-sided dice. Does anyone in the class have experience playing with any of these? If so, does the ELC seem reasonable to you? 7. A CM has a sample space that consists of four elements, denoted: a, b, c and d. Assuming the ELC, find the probabilities of each of the following events. (a) A = {a}

(b) B = {a, b}

(c) C = {b, c, d}

8. Refer to the previous problem. Now, instead of the ELC, assume that the probabilities of a, b, c and d follow the ratio 9:3:3:1. (a) Determine the probabilities of the individual outcomes a, b, c and d. 1

Assume i.i.d. trials. The following table helps to visualize the results of the first two trials:

Chapter 2 Lecture Examples 1. Consider a sample space with three members: 1, 2 and 3. Assume the ELC and i.i.d. trials. The following table helps to visualize the results of the first two trials:

X1 1 2 3

X2 1 2 (1,1) (1,2) (2,1) (2,2) (3,1) (3,2)

X1 1 2 3

3 (1,3) (2,3) (3,3)

Note that these nine entries are not equally likely. Define X = X1 + X2 , the total of the numbers obtained in the first two trials. Find the sampling distribution of X.

The nine entries in this table are equally likely. Define X = X1 + X2 , the total of the numbers obtained in the first two trials. Find the sampling distribution of X.

4. I cast my blue round-cornered die 1,000 times and obtained the following frequencies:

2. Consider a sample space with five members: 0, 1, 2, 3 and 4. Assume the ELC and i.i.d. trials. The following table helps to visualize the results of the first two trials:

X1 0 1 2 3 4

0 (0,0) (0,1) (0,2) (0,3) (0,4)

1 (0,1) (1,1) (2,1) (3,1) (4,1)

X2 2 (0,2) (1,2) (2,2) (3,2) (4,2)

3 (0,3) (1,3) (2,3) (3,3) (4,3)

1 (1,1) (2,1) (3,1)

X2 2 3 (1,2) (1,3) (2,2) (2,3) (3,2) (3,3)

212, 148, 109, 140, 152, and 239 for the outcomes 1, 2, 3, 4, 5 and 6, respectively. Based on the LLN, do you think the ELC applies?

4 (0,4) (1,4) (2,4) (3,4) (4,4)

5. Anna likes to play basketball. Assume that Anna’s free throw attempts are BT with p = 0.65.

The 25 entries in this table are equally likely.

(a) Anna will shoot three free throws. Calculate the probability that she obtains S, F, S, in that order.

Define X = X1 X2 , the product of the numbers obtained in the first two trials. Find the sampling distribution of X.

(b) Anna will shoot four free throws. Calculate the probability that she obtains S, S, F, S, in that order. (c) Anna will shoot five free throws. Calculate the probability that she obtains a total of exactly four successes.

3. Consider a sample space with three members: 1, 2 and 3. Do not assume the ELC. Instead assume the following: P (1) = 0.2, P (2) = 0.3 and P (3) = 0.5. 2

(b) P (X = 125). (0.0399)

Chap. 2 Lecture Examples: Cont.

(c) P (X ≤ 140). (0.9379)

(d) Next week, Anna will shoot five free throws on each of four days: Monday thru Thursday. If she makes exactly four free throws on a particular day, we say that the even ‘Brad’ has occurred.

(d) P (115 ≤ X ≤ 140). (0.7913) (e) P (118 ≤ X < 137). (0.6463)

i. Calculate the probability that Brad will occur on Monday and Tuesday and not occur on Wednesday and Thursday. (I am asking for one answer.) ii. Calculate the probability that Brad will occur a total of exactly two times next week.

8. In the 2008 presidential election in Wisconsin, Barack Obama received 1,677,211 votes and John McCain received 1,262,393 votes. In this example, I will ignore votes cast for any other candidates. (Eat your heart out, Ralph Nader.) The finite population size is N = 1,677,211 + 1,262,393 = 2,939,604. I will designate a vote for Obama as a success, giving p = 0.571 and q = 0.429.

(e) Next week Anna will shoot four free throws on Friday. The number of free throws she shoots on Saturday will equal the number that she makes on Friday. Let Y denote the total number of free throws that Anna makes on Friday and Saturday combined.

Imagine a lazy pollster named Larry. Larry plans to select n = 5 persons at random with replacement from the population. He counts the number of successes in his sample and calls it X. He decides that if X ≥ 3, then he will declare Obama to be the winner. If X ≤ 2, then he will declare McCain the winner. What is the probability that Larry will correctly predict the winner?

i. Calculate P (Y = 2). ii. Calculate P (Y = 6). (f) Next Sunday Anna will shoot seven free throws on Sunday. Define A to be the event that she makes a total of exactly five free throws. Let B be the event that she makes her first two free throws. Calculate P (AB).

9. Refer to the previous exercise. Larry decides that the answer we obtained, 0.6313, is too small. So he repeats the above with n =601 instead of n = 5. He will declare Obama the winner if X ≥ 301. What is the probability that Larry will correctly predict the winner?

6. Let X ∼ Bin(256, 0.50). Calculate the mean, variance and standard deviation of X. What is the formula for Z? 7. Let X ∼ Bin(625, 0.20). Use the snc to approximate the following probabilities. (The exact probabilities are inside the parentheses.) (a) P (X ≥ 130). (0.3235) 3

Chapter 3 Lecture Examples

8. Using the snc approximation, Bob constructs a 90% CI for p and gets 0.2000 ± 0.0294. Obtain the 98% CI for p for Bob’s data.

1. Nancy performs n = 300 BTs and obtains a total of 72 successes. Analyze Nancy’s data with 80%, 90% and 95% CIs. Compare your answers. Use the snc approximation. 2. Refer to the previous question. Suppose that Nancy performs n = 600 BTs and obtains a total of 144 successes. Analyze these new data with the 95% CI. Use the snc approximation. Compare this current answer to the answer in the previous question. 3. Use the website to obtain the exact twosided CI for p for the following situations. (a) n = 20; x = 5; 95%. (b) n = 24; x = 11; 95%. (c) n = 28; x = 12; 90%. 4. Use the website to obtain the exact onesided upper CI for p for the following situations. (a) n = 20; x = 2; 95%. (b) n = 33; x = 1; 95%. (c) n = 38; x = 3; 90%. 5. Tom wants to kill mosquitoes. He exposes 100 mosquitoes to a certain poison and 72 of them die. How would you analyze these data? 6. Molly examines 200 plants of a certain type. She notes that four of the plants are diseased. How would you analyze these data? 7. Discuss the meaning of 95% confidence. 4

Chapter 4 Lecture Examples

7. Refer to the previous question. Suppose that Tammy and Ralph observe a total of 17 successes. Now, pretending that we don’t know that the rate is 3 per hour, use the website to calculate the 95% CI for the rate. Is your CI correct?

1. Suppose that X ∼ Poisson(16). Calculate the mean, variance and standard deviation of X. 2. Suppose that X ∼ Poisson(16). Use the website to calculate the following probabilities. (a) P (X = 15). (b) P (X ≤ 15). (c) P (X > 15).

(d) P (16 ≤ X ≤ 24). 3. Suppose that X ∼ Poisson(100). Use the snc to approximate the following probabilities. Also, use the website to obtain the exact probabilities. (a) P (X > 105). (b) P (90 < X < 116). 4. Suppose that X ∼ Poisson(θ), with θ unknown. Given that X = 80, (a) Use the snc to obtain the approximate 90% CI for θ. (b) Use the website to obtain the exact 90% CI for θ. 5. Tammy is observing a Poisson Process with a rate of 3 occurrences per hour. Let X denote the number of successes she will observe in a 2.5 hour period. Use the website to find P (X = 9). 6. Refer to the previous question. After Tammy has finished her data collection, Ralph observes the same process for 90 minutes. Let Y denote the total number of successes observed by Tammy and Ralph. Use the website to find P (Y ≤ 15). 5

7. Bob shoots m = 3 free throws every day for n = 160 days. Each day he counts the number of successes he obtains. His data are below.

Chapter 5 Lecture Examples 1. Describe how to find the area under the χ2 (6) to the right of 8.04. 2. Describe how to find the χ20.01 (6). The website gives me 16.81; describe what this means.

Number of S’s 0 Number of days 44

Are these data consistent with Bob’s shots being BT with p unknown? Use α = 0.01. Also, calculate the P-value.

3. Suppose that a male human with AO blood impregnates a female human with BO blood. (a) What are the possible blood types for the child? (b) Do you have hypothesized probabilities for each of these blood types? 4. Discuss Type 1 and Type 2 errors in the context of the snapdragon example in the Course Notes. 5. (Hypothetical data.) A cross between white and yellow summer squash gave the following colors. Color White Yellow Green Number 244 50 26 Are these data consistent with the 12:3:1 ratio predicted by a genetic model? Use α = 0.05. Also, calculate the P-value. 6. (Hypothetical data.) In a breeding experiment, white chickens with small combs were mated and produced 240 offspring. Are these data consistent with the 9:3:3:1 ratio predicted by a genetic model? Use α = 0.10. Also, calculate the P-value. White feathers Type Small comb Number 129

1 2 3 40 34 42

White feathers Large comb 47

6

Dark feathers Small comb 47

Dark feathers Large comb 17

(b) Investigate the second assumption of BT using two segments of size 48.

Chapter 6 Lecture Examples 1. If we divide Katie’s data (Model Project 5) into four equal sized segments of 25 trials, her number of successes per segment are: 6, 8, 5 and 10.

(c) Investigate the issue of independence of trials. 3. Below is a listing of Boone’s data. Use these data to check for memory.

Analyze these data descriptively and also with a Chi-Squared test.

S F F S S S F S F S

2. In her report on her statistics project, Kristen Joiner wrote:

S F S S S F S S S S I chose to test Muffin’s preference for balls. She has two balls we often throw for her to fetch. One is small and blue, the other is slightly larger, heavier and red. My father rolled both balls for her with one hand at the same time; I recorded which ball she chose to chase first. Arbitrarily, the blue ball was labeled a success.

F S S F S S S F S S F F S S S S S S S F F S F S S F S F F S S F S F S F S S S F S F S S F S S S S S S S S S S S S S S F

Here is a listing of Kristen’s 96 trials, in order from left to right.

S F S S S S F S F F

S

S

S

F

F

S

S

S

S

F

F

S

S

F

S

F

F

S

F

S

S

S

S

F

S

S

S

F

F

F

S

S

S

S

F

F

F

S

F

F

S

F

S

F

S

F

S

F

F

F

F

F

F

S

F

S

F

F

S

S

S

S

F

S

S

F

S

F

S

S

S

S

S

F

F

S

S

F

F

S

S

S

S

S

S

S

F

S

S

S

S

S

S

F

F

S

(a) Investigate the second assumption of BT using four segments of size 24. 7

S S S S S S F S S S 4.

(a) Below are tables (1–3) from three different studies. Match each Table to the correct statement below. A This table provides evidence that p increased over the course of the study. B This table provides evidence that p decreased over the course of the study. C This table provides no evidence that p changed over the course of the study.

Prev. S F Total

Table 4 Current S F 96 144 144 216 240 360

Total 240 360 600

Prev. S F Total

Table 5 Current S F 77 78 77 122 154 200

Total 155 199 354

Prev. S F Total

Table 6 Current S F 82 74 75 36 157 110

Total 156 111 267

Chap. 6 Lecture Examples: Cont.

Half 1st 2nd

Table 1 S F 54 89 65 78

Half 1st 2nd

Table 2 S F 124 93 124 93

Total 217 217

Half 1st 2nd

Table 3 S F 85 135 75 145

Total 220 220

Total 143 143

5. Below are tables (4, 5 and 6) from three different studies. Note that at least one of these tables is the answer to more than one of the following six questions.

6. Discuss the turtle study.

(a) Which table is for the study that had an F on its first trial and an S on its last trial? (b) Which table is for the study that had an F on its first and last trials? (c) Which table is for the study that had an S on its first trial and an F on its last trial? (d) Which table is for the study in which the majority of trials yielded an S? (e) Which table is for the study in which the proportion of successes after an S was smaller than the proportion of successes after an F? (f) Which table is for the study in which the proportion of successes after an S was equal to the proportion of successes after an F? 8

(e) After the trials are observed, it is discovered that y = 150 successes were obtained. Comment on your answers in (a) and (c).

Chapter 7 Lecture Examples 1. Alma plans to observe Y ∼ Bin(200,0.40). (a) Calculate the point prediction yˆ.

4. Dave plans to observe Poisson Process with unknown rate λ per minute. He plans to observe the PP for two hours and uses Y to denote the total number of successes that he will observe. Dave has previous data on the PP for which 80 minutes yielded 294 successes.

(b) Use the binomial calculator to determine the probability that the point prediction will be correct. (c) Use the snc approximation to obtain the 90% PI for Y . (d) Use the binomial calculator to determine the probability that the PI will be correct. Should it be changed?

(a) Calculate the point prediction yˆ. (b) Use the snc approximation to obtain the 80% PI for Y .

(e) After the trials are observed, it is discovered that y = 95 successes were obtained. Comment on your answers in (a) and (c).

(c) After the trials are observed, it is discovered that y = 470 successes were obtained. Comment on your answers in (a) and (b).

2. Beth plans to observe m = 300 BT. She does not know the value of p. Beth has previous data which consists of x = 62 successes in n = 200 BT. (a) Calculate the point prediction yˆ. (b) Obtain the 90% PI for Y . (c) After the trials are observed, it is discovered that y = 100 successes were obtained. Comment on your answers in (a) and (b). 3. Carly plans to observe Y ∼ Poisson(169). (a) Calculate the point prediction yˆ. (b) Use the Poisson calculator to determine the probability that the point prediction will be correct. (c) Use the snc approximation to obtain the 80% PI for Y . (d) Use the Poisson calculator to determine the probability that the PI will be correct. Should it be changed? 9

Chapter 9 Lecture Examples 1. (This is a real study. See my textbook for a reference to it.) In 1986, 129 women who visited a gynecologist at a university health service identified themselves as ‘sexually experienced.’ When asked whether their partner uses a condom during sexual intercourse, 30 gave the answer ‘Always or almost always;’ the other choice was ‘Seldom or never.’ (It is a pretty big gap between seldom and almost always.) Identifying the first of these choices as a success, use the snc approximation to calculate the 95% CI for p. What is the population? Was it a random sample? If not, do you think it matters? 2. Refer to the previous question. In 1989 at the same university, same situation, in a sample of 112 women there were 46 successes. Use the snc approximation to calculate the 95% CI for p. Compare this answer to the answer to question 1 and comment. 3. Refer to the previous two questions. Identify 1989 as population 1 and 1986 as population 2. (a) Calculate the 95% CI for p1 −p2 . Discuss your answer. (b) Perform the test of hypotheses for comparing the two populations. Use α = 0.05 and the alternative p1 > p2 . Use the snc approximation and the exact website. Discuss your answers. 4. For the 2008–2009 men’s basketball team, Joe Krabbenhoft made 66 free throws in 78 attempts. Marcus Landry made 57 free throws in 91 attempts. 10

Use these data to decide which man was the better free throw shooter. Use both a confidence interval and a test of hypotheses. For the test use the two-sided alternative and α = 0.05. (To greatly increase the chance we all get the same answer, take Joe’s shots to be population 1.) 5. An observational study yields the following “collapsed table.” Group 1 2 Total

S F Total 72 228 300 88 212 300 160 440 600

Below are two component tables for these data. Complete these tables so that Simpson’s Paradox is occurring or explain why Simpson’s Paradox cannot occur for these data. For the latter, you must provide computations that justify your answer. Subgp A Subgp B Gp S F Tot Gp S F Tot 1 30 30 60 1 42 198 240 2 120 2 180 Tot 180 Tot 420 6. During the 1981–82 NHL season, Wayne Gretzky scored 212 points in 80 games. (In hockey, a player scores one point if he/she either scores a goal or assists on a goal.) In the next season, he scored 196 points in 80 games. Assume that the number of points Gretzky would score followed a PP with rate λ1 points per game in the earlier season and λ2 , points per game in the latter season. (a) Calculate the 95% CI for λ1 − λ2 . (b) Find the P-value for testing the null that λ1 = λ2 , versus the two-sided alternative.

Chapter 10 Lecture Examples 1. I have drawn a histogram for 400 observations. One of the rectangles has endpoints of 0.50 and 0.60, and a height of 0.25. For each of the three situations below, determine how many observations are in this class interval [0.50 to 0.60), with the usual endpoint convention.

(b) If it is a relative frequency histogram. (c) If it is a density scale histogram. 2. Below are 100 sorted observations. 0.02 0.14 0.16 0.31 0.40 0.51 0.61 0.82 0.99 1.19 1.39 1.59 1.79 1.97 2.23 2.56 3.37 3.85 4.52 5.48

0.02 0.14 0.20 0.32 0.47 0.53 0.62 0.86 1.03 1.29 1.46 1.61 1.79 2.09 2.38 2.71 3.49 4.05 4.74 5.90

0.05 0.15 0.21 0.34 0.47 0.53 0.63 0.93 1.04 1.35 1.48 1.70 1.79 2.11 2.39 2.88 3.56 4.20 5.16 6.75

3. Refer to the data in example 2 above. (a) Calculate the first and third quartiles of these data.

(a) If it is a frequency histogram.

0.01 0.09 0.15 0.26 0.39 0.49 0.59 0.78 0.97 1.18 1.37 1.52 1.75 1.86 2.16 2.52 3.13 3.61 4.40 5.37

(c) Draw a density scale histogram of these data. Use 0.00–0.50, 0.50–1.00, 1.00–2.00, 2.00–4.00 and 4.00–7.00 as your class intervals. Clearly label the height and endpoints of each of the four rectangles.

0.06 0.15 0.24 0.38 0.49 0.55 0.64 0.94 1.05 1.36 1.51 1.70 1.81 2.13 2.49 3.00 3.58 4.22 5.27 6.82

(b) Given the mean and standard deviation of these data are 1.78 and 1.65, respectively, determine the proportion of observations that are within one standard deviation of the mean. How does your proportion compare to value predicted by the empirical rule? Are you are surprised by the agreement/disagreement? Comment. 4. A sample of size n = 28 yields the following sorted data. Note that the largest number in the list has been replaced by the letter ‘y.’ 384 492 594 650

420 494 618 686

422 518 622 712

456 520 630 754

462 572 642 764

466 486 576 580 644 646 790 y

Hint: The mean of these data is 586.0. (a) Calculate the median and the first quartile of these data.

(a) Calculate the median of these data. (b) Draw a relative frequency histogram of these data. Use five intervals of equal width, beginning at 0.00 and ending at 7.50. Clearly label the height and endpoints of each of the five rectangles. 11

(b) Suppose we discover that the observation 384 is an error. Recalculate the median, the first quartile and the mean after deleting the observation 384.

0.325, 0.357, 0.387, 0.404, 0.422,

Chapter 11 Lecture Examples 1. The cat population has the following probability distribution. x P (X = x)

0 1 2 3 0.1 0.5 0.3 0.1

It can be shown that µ = 1.4 and σ = 0.8. For n = 25, using√Formula (11.2) we get that x¯ ± 1.96(0.8/ 25) = x¯ ± 0.3136 is an approximate 95% CI for µ, when σ is known. I performed a 10,000 run simulation study on this CI. (a) Explain, in detail, the steps in each run.

0.439, 0.506, 0.540. Use these data to calculate a CI for the population median, ν. Use a confidence level that is close to 95%. 3. Refer to the problem 1. (a) I performed a 10,000 run simulation study with 95% confidence level, n = 10 and both the Slutsky (z = 1.96) and Gosset (t = 2.262) CI’s for the cat population. Here is what I found: • For Slutsky, 451 CI’s were too small and 371 were too large. • For Gosset, 362 CI’s were too small and 148 were too large.

(b) Here are the results of the simulation study: 268 simulated CI’s were too small (this means that the upper bound of the CI was smaller than µ = 1.4); and 340 simulated CI’s were too large (this means that the lower bound of the CI was larger than µ = 1.4). Discuss these findings.

(b) I performed a 10,000 run simulation study with 95% confidence level, n = 20 and both the Slutsky (z = 1.96) and Gosset (t = 2.093) CI’s for the cat population. Here is what I found:

2. A researcher obtained the weights, in grams, of n = 13 web weaver spiders. We have the following summary statistics: x¯ = 0.3582 and s = 0.1140.

• For Slutsky, 377 CI’s were too small and 286 were too large. • For Gosset, 332 CI’s were too small and 208 were too large.

(a) Calculate Gosset’s 95% CI for the population mean weight of web weaver spiders. (b) How would you obtain a random sample of web weaver spiders? (c) Test the null hypothesis that the population mean equals 0.5 grams versus the two-sided alternative. Find the Pvalue and use α = 0.05. (d) Here are the sorted weights of these spiders: 0.106, 0.234, 0.287, 0.324, 0.325, 12

Discuss these findings.

Discuss these findings. (c) I performed a 10,000 run simulation study with 95% confidence level, n = 40 and both the Slutsky (z = 1.96) and Gosset (t = 2.023) CI’s for the cat population. Here is what I found: • For Slutsky, 363 CI’s were too small and 245 were too large. • For Gosset, 320 CI’s were too small and were 209 too large. Discuss these findings.

Chap. 11 Lecture Examples: Cont. 4. On page 526 of my textbook is a picture of the lognormal pdf with parameters 5 and 1. (This means that if we begin with Y which has a N(5,1) pdf, then X = eY has this lognormal pdf.) It is suffice to note that X must be positive and its pdf is strongly skewed to the right, as evidenced by µ = 244.7 begin about 65% larger than ν = 148.4. Another indication of skewness is that, even though X cannot be negative, σ = 320.8 is larger than µ = 244.7. When I was writing my text, I performed several 5,000 run simulation studies to see how Gosset’s CI worked for this lognormal and various choices of n. My results are summarized in the table below. % of Correct n ‘95%’ Gosset CI’s 10 0.8410 20 0.8764 30 0.8858 100 0.9092 150 0.9254 200 0.9306 Briefly discuss what this table reveals.

13

Chapter 12 Lecture Examples 1. A 1991 study compared a treatment and a placebo in postmenopausal women with osteoporosis. (See the references to Chapter 16 of my text for more information.) Forty subjects were available for study and they were divided into two equal sized groups by randomization. The response was the percentage change in ‘bone mineral content’ from baseline to the end of the study. In the first group, calcium supplement, the sample mean was 5.3 percent with a standard deviation of 7.05 percent. In the second group, placebo, the sample mean was −2.7 percent with a standard deviation of 9.83 percent. (a) Use Case 3 to obtain the 95% CI for µ1 − µ2 .

(b) Use Case 1 to obtain the 95% CI for µ1 − µ2 . (c) Compare your answers in (a) and (b).

(d) Find the P-value for testing µ1 = µ2 versus µ1 > µ2 . Use Case 3. (e) Repeat (d) for Case 1. (f) Compare your answers in (d) and (e). 2. Calculate sp in each of the following cases. (a) n1 = 8, n2 = 14, s1 = 5.00 and s2 = 8.50. (b) n1 = 12, n2 = 15, s1 = 9.00 and s2 = 11.50. 3. For this problem, refer to the t-curve calculator linked to our course website. Remember that there are three boxes and a choice. The top box is for the df . The left box has a choice, either ‘Area left of’ and ‘Area right of.’ For each of the situations below, first tell me what to enter in each box and which 14

choice to select to obtain the desired answer. Next, do it and tell me the answer. (a) n1 = 8, n2 = 14 and we want the t needed for the 90% CI for µ1 − µ2 .

(b) n1 = 12, n2 = 15 and we want the t needed for the 98% CI for µ1 − µ2 . (c) n1 = 8, n2 = 14, t = 2.734 and we want the P-value for the alternative >.

(d) n1 = 8, n2 = 14, t = 2.734 and we want the P-value for the alternative 6=. (e) n1 = 12, n2 = 15, t = −1.342 and we want the P-value for the alternative 25. Season Location 1967 1968 1969 1970 Home 160 166 148 201 Away 110 102 112 121 H−A 50 64 36 80 1971 1972 1973 1974 1975 1976 144 146 138 139 125 155 116 99 107 93 100 73 28 47 31 46 25 82 1977 1978 1979 1980 1982 151 117 151 116 115 88 80 111 100 112 63 57 40 16 3 1983 1984 1985 1986 1987 140 156 202 168 204 117 79 104 130 164 23 77 98 38 40

15

(d) On a given shot, given that the dog barks, what is the probability my neighbor made the shot?

Chapter 13 Lecture Examples 1. Below is a table of population counts. A Ac Total

B 60 140 200

Bc 40 260 300

Total 100 400 500

(e) On a given shot, given that the dog does not bark, what is the probability my neighbor missed the shot?

(a) Create the table of population proportions, a.k.a. probabilities.

3. Below are data that you have seen previously in homework. They are the data on Rick Robey shooting free throws.

(b) Create the table of the conditional probabilities of the A’s given the B’s. (c) Create the table of the conditional probabilities of the B’s given the A’s. (d) Suppose that this table is for a screening test for a disease. Describe, in words, each of the eight conditional probabilities. 2. My neighbor has trained her dog to bark whenever she (my neighbor, not the dog) makes a basket. From my perch on my porch, I can see my neighbor shooting free throws, but cannot see the basket she is shooting at. In addition, whether she makes the shot or not, she gives no reaction that I can see. In addition, she claims all of the following to be true: • Her free throws are BT’s with p = 0.75. • Given that she makes (misses) a free throw, there is an 80% (28%) chance that her dog will bark. Assume that all of my neighbors probabilities are correct. (a) Define events A, Ac , B and B c

First Shot Made (A) Missed (Ac ) Total

Second Shot Made (B) Missed (B c ) 54 37 49 31 103 68

(a) Find the P-values for the test of pA = pB versus each of the three possible alternatives. (b) Obtain the 80% CI for pA − pB . 4. Linda performed a randomized pairs design with 50 trials to investigate whether Maisie is better at catching a 28 inch frisbee (the first treatment) or a 36 inch frisbee in her mouth. Each trial was a toss of a frisbee and Maisie was credited with a success if she caught the frisbee before it hit the ground. Here are Maisie’s results: • For six pairs of trials she obtained a success with each frisbee. • For four pairs of trials she obtained a failure with each frisbee. • For six pairs of trials she obtained a success with the smaller frisbee but a failure with the larger frisbee.

(b) Create the table of probabilities. (c) Calculate the probability that my neighbor’s dog will bark on any given shot. 16

Total 91 80 171

Answer questions (a) and (b) from 3.

Chapter 14 Lecture Examples Use the following computer output to answer questions 1–11. Regression Analysis: Runs versus OPS The regression equation is Runs = - 912 + 2205 OPS Predictor Constant OPS S = 16.91

Coef -912.4 2205.4

SE Coef 120.5 163.0

R-Sq = 92.9%

T -7.57 13.53

P 0.000 0.000

R-Sq(adj) = 92.4%

Analysis of Variance Source Regression Residual Error Total Obs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

OPS 0.781 0.784 0.767 0.758 0.756 0.744 0.747 0.742 0.743 0.738 0.712 0.729 0.699 0.719 0.701 0.705

DF 1 14 15 Runs 820.00 804.00 785.00 780.00 772.00 735.00 730.00 720.00 710.00 707.00 673.00 671.00 657.00 643.00 638.00 636.00

SS 52368 4004 56372 Fit 810.05 816.67 779.18 759.33 754.92 728.45 735.07 724.04 726.25 715.22 657.88 695.37 629.21 673.32 633.62 642.44

17

MS 52368 286

SE Fit 8.04 8.46 6.21 5.23 5.05 4.30 4.42 4.25 4.28 4.23 6.11 4.53 7.78 5.34 7.51 6.98

F 183.12

Residual 9.95 -12.67 5.82 20.67 17.08 6.55 -5.07 -4.04 -16.25 -8.22 15.12 -24.37 27.79 -30.32 4.38 -6.44

P 0.000

St Resid 0.67 -0.87 0.37 1.29 1.06 0.40 -0.31 -0.25 -0.99 -0.50 0.96 -1.50 1.85 -1.89 0.29 -0.42

Chap. 14 Lecture Examples: Cont. 1. What is the point estimate of the slope of the regression line? Intepret this number.

12. This question is not about the prediction line; instead, it is about the Principle of Least Squares. You are given the following data: x: y:

2. Calculate the 99% CI for the slope of the regression line.

0 1 2 1 3 5

3 9

3. Find the P-value for testing the null hypothesis that β1 = 2000 versus the alternative β1 > 2000.

There are two candidates for prediction:

4. Calculate the 98% CI for the mean of Y given that X = 0.781.

According to the Principle of Least Squares, which prediction rule is better?

5. Calculate the 98% CI for the mean of Y given that X = 0.738.

13. For the least squares line, remember that there are two restrictions on the residuals, P P e = 0 and (xe) = 0. Determine the values of e3 and e4 in the following table.

6. Calculate the 90% prediction interval for the value of Y given that X = 0.767. 7. Calculate the 90% prediction interval for the value of Y given that X = 0.701. 8. The output tells you that s = 16.91. Calculate this value by using the ANOVA table output. 9. The output tells you that R2 = 0.929. Calculate this value by using the ANOVA table output. 10. Fill in the blanks in the following. According to the empirical rule, about 68% of the residuals will fall between and . Determine, by counting and then dividing by 16, the actual percentage of residuals that fall between your two numbers above. 11. You are given that the mean of the 16 OPS values is 0.73906. Look at how SE Fit varies with the value of OPS; do you see a pattern? 18

y˙ = x2 + 1 and y¨ = 2x + 1.

x: e:

1 2 3 4 5 1 −3 e3 e4 −2