Chapter 1 Homework

Recall the previous exercise on cardiac arrests for kidney disease patients. Besides the Monday-Wednesday-Friday schedule for dialysis, many patients are on a Tuesday-Thursday-Saturday schedule. Again, like Monday, Tuesday is unique in that there is the same preceding two-day gap in dialysis. The researchers found that of the 190 cardiac arrests that took place for patients on the Tuesday-Thursday-Saturday schedule, 65 of them occurred on Tuesdays. Do we have convincing evidence of a larger probability of cardiac arrests occurring on Tuesdays compared to the other two days? Investigate by answering the following. H0: HA: Give the value of the statistic. a. p̂ = 0.333 b. π = 0.342 c. π = 0.333 d. p̂ = 0.342 Are the validity conditions met? Explain. a. No, because the number of success (65) and failures (125) are each at least 10. b. Yes, because the sample size is more than 20. c. Yes, because the number of success (65) and failures (125) are each at least 10. d. No, because the sample size is more than 20.

1) H0: π=0.333 HA: π>0.333 2) d/p̂ = 0.342 3) c/Yes, because the number of success (65) and failures (125) are each at least 10.

Patients with kidney disease often have a procedure called dialysis done to clean their blood if their kidneys can't do this properly. This procedure is often done three days per week, with Monday, Wednesday, and Friday often being those days. In terms of the dialysis treatments, these days are all the same except for the gap of two off days before the Monday treatment. Does this gap make a difference? Cardiac arrest and sudden death for patients undergoing dialysis for kidney disease are possibilities. In recent years, it has been noted that these types of patients have cardiac arrests on Mondays more often than what would be expected. A study published in Kidney International (Karnik et al., 2001) looked at 205 dialysis patients that had cardiac arrests on Monday, Wednesday, or Friday, the same days they had dialysis. They found that 93 of these happened on a Monday. Do we have convincing evidence of a larger probability of cardiac arrests occurring on Mondays compared to the other two days? Investigate by answering the following. 1) Use the One Proportion applet to report a simulation-based p-value. Choose the best among the following options a. < 0.001 b. 0.528 c. 0.472 d. 1 2) Use the One Proportion applet to also report a theory-based p-value. a. 0.0001 b. 1 c. 0.0002 d. 0.5000

1) a/< 0.001 2) a/0.0001

Recall the previous exercise on cardiac arrests for kidney disease patients. Besides the Monday-Wednesday-Friday schedule for dialysis, many patients are on a Tuesday-Thursday-Saturday schedule. Again, like Monday, Tuesday is unique in that there is the same preceding two-day gap in dialysis. The researchers found that of the 190 cardiac arrests that took place for patients on the Tuesday-Thursday-Saturday schedule, 65 of them occurred on Tuesdays. Do we have convincing evidence of a larger probability of cardiac arrests occurring on Tuesdays compared to the other two days? Investigate by answering the following. 1) Use the One Proportion applet to report a simulation-based p-value. Choose the best among the following options a. 0.001 b. 1 c. 0.415 d. 0.826 2) Use the One Proportion applet to also report a theory-based p-value. a. 0.7924 b. 0.0014 c. 1 d. 0.3962

1) c/0.415 2) d/0.3962

Patients with kidney disease often have a procedure called dialysis done to clean their blood if their kidneys can't do this properly. This procedure is often done three days per week, with Monday, Wednesday, and Friday often being those days. In terms of the dialysis treatments, these days are all the same except for the gap of two off days before the Monday treatment. Does this gap make a difference? Cardiac arrest and sudden death for patients undergoing dialysis for kidney disease are possibilities. In recent years, it has been noted that these types of patients have cardiac arrests on Mondays more often than what would be expected. A study published in Kidney International (Karnik et al., 2001) looked at 205 dialysis patients that had cardiac arrests on Monday, Wednesday, or Friday, the same days they had dialysis. They found that 93 of these happened on a Monday. Do we have convincing evidence of a larger probability of cardiac arrests occurring on Mondays compared to the other two days? Investigate by answering the following. 1) Give the value of the statistic. a. p̂ = 0.500 b. p̂ = 0.454 c. π = 0.500 d. π = 0.454 2) Are the validity conditions met? Explain. a. Yes, because the number of success (112) and failures (93) are each at least 10. b. No, because the number of success (112) and failures (93) are each at least 10. c. No, because the number of success (93) and failures (112) are each at least 10. d. Yes, because the number of success (93) and failures (112) are each at least 10.

1. b/p̂ = 0.454 2. d/Yes, because the number of success (93) and failures (112) are each at least 10.

Patients with kidney disease often have a procedure called dialysis done to clean their blood if their kidneys can't do this properly. This procedure is often done three days per week, with Monday, Wednesday, and Friday often being those days. In terms of the dialysis treatments, these days are all the same except for the gap of two off days before the Monday treatment. Does this gap make a difference? Cardiac arrest and sudden death for patients undergoing dialysis for kidney disease are possibilities. In recent years, it has been noted that these types of patients have cardiac arrests on Mondays more often than what would be expected. A study published in Kidney International (Karnik et al., 2001) looked at 205 dialysis patients that had cardiac arrests on Monday, Wednesday, or Friday, the same days they had dialysis. They found that 93 of these happened on a Monday. Do we have convincing evidence of a larger probability of cardiac arrests occurring on Mondays compared to the other two days? Investigate by answering the following. Set up the correct null and alternative hypotheses in symbols. H0: HA+

H0: π=0.333 HA: π>0.333

Student researchers were interested in whether people will be more likely to choose the name-brand cookie (Chips Ahoy) over the store-brand (Chipsters) in a blind taste test. They tested this with 20 subjects and found that 14 (70%) chose Chips Ahoy as their favorite. They conducted a test of significance using simulation and got the following null distribution. (Note that this null distribution uses only 100 simulated samples and not the usual 1,000 or 10,000.) We will let π represent the long-run proportion of students that pick Chips Ahoy as their favorite. a. Set up the correct the null and alternative hypotheses in symbols for the research question, "Do a majority of students prefer Chips Ahoy over Chipsters?" H0= HA=

H0: π=0.5 HA: π>0.5

Student researchers were interested in whether people will be more likely to choose the name-brand cookie (Chips Ahoy) over the store-brand (Chipsters) in a blind taste test. They tested this with 20 subjects and found that 14 (70%) chose Chips Ahoy as their favorite. They conducted a test of significance using simulation and got the following null distribution. (Note that this null distribution uses only 100 simulated samples and not the usual 1,000 or 10,000.) We will let π represent the long-run proportion of students that pick Chips Ahoy as their favorite. Set up the correct the null and alternative hypotheses in symbols for the research question, "Do students have a preference between Chips Ahoy and Chipsters?" H0: HA:

H0: π=0.50 HA: π ≠ 0.50

Which sample size, n, gives the smallest standard deviation of the null distribution where the long-run proportion, π, is 0.25? Select one: A. 60 B. 30 C. 50 D. 40

a/60

Suppose you are testing the hypotheses H0: π = 0.50 versus Ha: π > 0.50. You get a sample proportion of 0.68 and find that your p-value is 0.02. Now suppose you redid your study with each of the following changes, will your new p-value be larger, smaller, or stay the same as the 0.02 you first obtained? Be sure to explain your reasoning. a. Keeping the sample size the same, you take a new sample and find a sample proportion of 0.66. The new p-value will be *larger or smaller* because *option 1, 2, or 3* b. You increase the sample size and still find a sample proportion of 0.68. The new p-value will be *larger or smaller* because *option 1, 2, or 3* c. You decide to use a two-sided alternative hypothesis (Ha: π ≠ 0.50). The new p-value will be *larger or smaller* because *option 1, 2, or 3* Option 1: the p-value calculation now includes at least as extreme values of the statistic in both tails of the null distribution Option 2: the sample proportion is closer to the hypothesized long-run proportion value of 0.5 Option 3: it is less likely to get extreme value of the statistic from a larger sample

a. larger, option 2 b. smaller, option 3 c. larger, option 1

Suppose you are testing the hypotheses H0: π = 0.50 versus Ha: π > 0.50. You get a sample proportion of 0.68 and find that your standardized statistic is 2.53. Now suppose you redid your study with each of the following changes; will your new standardized statistic be larger, smaller, or stay the same as the 2.53 you first obtained? Explain your reasoning. a. Keeping the sample size the same, you take a new sample and find a sample proportion of 0.66. The new standardized statistic will be *smaller (closer to 0) 0r the same or larger (farther from 0)* because *option 1, 2, or 3* b. You increase the sample size and still find a sample proportion of 0.68. The new standardized statistic will be *smaller (closer to 0) 0r the same or larger (farther from 0)* because *option 1, 2, or 3* c. You decide to use a two-sided alternative hypothesis (Ha: π ≠ 0.50). The new standardized statistic will be *smaller (closer to 0) 0r the same or larger (farther from 0)* because *option 1, 2, or 3* Option 1: the sample proportion os closer to the hypothesizes long run proportion value of 0.50 Option 2: it is less likely to get extreme values of the statistic from a larger sample Option 3: the sample proportion is still the same distance away from the center of the null distribution

a. smaller (closer to 0), option 1 b. larger (farther from 0), option 2 c. the same, option 3

Recall the previous exercise on cardiac arrests for kidney disease patients. Besides the Monday-Wednesday-Friday schedule for dialysis, many patients are on a Tuesday-Thursday-Saturday schedule. Again, like Monday, Tuesday is unique in that there is the same preceding two-day gap in dialysis. The researchers found that of the 190 cardiac arrests that took place for patients on the Tuesday-Thursday-Saturday schedule, 65 of them occurred on Tuesdays. Do we have convincing evidence of a larger probability of cardiac arrests occurring on Tuesdays compared to the other two days? Investigate by answering the following. Write out a conclusion in the context of the research question. a. Based on the large p-value, we do not have strong evidence against the null hypothesis. We do not have strong evidence that the long-run proportion of cardiac arrests among dialysis patients that happen on Tuesdays (compared to Thursdays or Saturdays) is greater than 0.333. b. Based on the small p-value, we have strong evidence against the null hypothesis. We have strong evidence that the long-run proportion of cardiac arrests among dialysis patients that happen on Tuesdays (compared to Thursdays or Saturdays) is greater than 0.333. c. Based on the large p-value, we have strong evidence against the null hypothesis. We do not have strong evidence that there is a larger probability of cardiac arrests among dialysis patients occurring on Tuesdays compared to Thursdays or Saturdays. d. Based on the small p-value, we do not have strong evidence against the null hypothesis. We have strong evidence that there is a larger probability of cardiac arrests among dialysis patients occurring on Tuesdays compared to Thursdays or Saturdays.

a/Based on the large p-value, we do not have strong evidence against the null hypothesis. We do not have strong evidence that the long-run proportion of cardiac arrests among dialysis patients that happen on Tuesdays (compared to Thursdays or Saturdays) is greater than 0.333.

Suppose a friend of yours says she is a 75% free-throw shooter in basketball. You don't think she is that good and want to test her to gather evidence that she makes less than 75% of her free throws in the long run. You have her shoot 30 free throws and she makes 18 (or 60%) of them. Are the validity conditions met for the one-proportion z-test? Select one: A.Yes, because the number of success and failures are each at least 10. B.No, because the sample size is less than 40. C.No, because the number of success and failures are each at least 10. D.Yes, because the sample size is more than 20.

a/Yes, because the number of success and failures are each at least 10.

Student researchers were interested in whether people will be more likely to choose the name-brand cookie (Chips Ahoy) over the store-brand (Chipsters) in a blind taste test. They tested this with 20 subjects and found that 14 (70%) chose Chips Ahoy as their favorite. They conducted a test of significance using simulation and got the following null distribution. (Note that this null distribution uses only 100 simulated samples and not the usual 1,000 or 10,000.) We will let π represent the long-run proportion of students that pick Chips Ahoy as their favorite. Using the null distribution above, what is the p-value when testing the hypotheses in part (c)? a. p-value = 8/100 = 0.08 b. p-value = 2/100 = 0.02 c. p-value = 6/100 = 0.06 d. p-value = 100/100 = 1

a/p-value = 8/100 = 0.08

Suppose you are using theory-based techniques (e.g., a one-proportion z-test) to determine p-values. How will a two-sided p-value compare to a one-sided p-value (assuming the one-sided p-value is less than 0.50)? Select one: A.The two-sided p-value will be close to twice as large as the one-sided. B.The two-sided p-value will be exactly twice as large as the one-sided. C.The two-sided p-value will be about the same as the one-sided. D.The two-sided p-value will be half as much as the one-sided.

b/The two-sided p-value will be exactly twice as large as the one-sided.

Patients with kidney disease often have a procedure called dialysis done to clean their blood if their kidneys can't do this properly. This procedure is often done three days per week, with Monday, Wednesday, and Friday often being those days. In terms of the dialysis treatments, these days are all the same except for the gap of two off days before the Monday treatment. Does this gap make a difference? Cardiac arrest and sudden death for patients undergoing dialysis for kidney disease are possibilities. In recent years, it has been noted that these types of patients have cardiac arrests on Mondays more often than what would be expected. A study published in Kidney International (Karnik et al., 2001) looked at 205 dialysis patients that had cardiac arrests on Monday, Wednesday, or Friday, the same days they had dialysis. They found that 93 of these happened on a Monday. Do we have convincing evidence of a larger probability of cardiac arrests occurring on Mondays compared to the other two days? Investigate by answering the following. Write out a conclusion in the context of the research question. a. Based on the large p-value, we have strong evidence against the null hypothesis. We do not have strong evidence that there is a larger probability of cardiac arrests among dialysis patients occurring on Mondays compared to Wednesdays or Fridays. b. Based on the small p-value, we do not have strong evidence against the null hypothesis. We have strong evidence that there is a larger probability of cardiac arrests among dialysis patients occurring on Mondays compared to Wednesdays or Fridays. c. Based on the small p-value, we have strong evidence against the null hypothesis. We have strong evidence that the long-run proportion of cardiac arrests among dialysis patients that happen on Mondays (compared to Wednesdays or Fridays) is greater than 0.333. d. Based on the large p-value, we do not have strong evidence against the null hypothesis. We do not have strong evidence that the long-run proportion of cardiac arrests among dialysis patients that happen on Mondays (compared to Wednesdays or Fridays) is greater than 0.333.

c/Based on the small p-value, we have strong evidence against the null hypothesis. We have strong evidence that the long-run proportion of cardiac arrests among dialysis patients that happen on Mondays (compared to Wednesdays or Fridays) is greater than 0.333.

The theorem that states that if the sample size is large enough, the distribution of sample proportions will be bell-shaped (approximately normal), centered at the long run proportion π, with a standard deviation of is called: √(π(1-π)/n) Select one: A. The theory-based theorem. B. The normal theorem. C. The central limit theorem. D. The fundamental theorem of statistics.

c/The central limit theorem.

Student researchers were interested in whether people will be more likely to choose the name-brand cookie (Chips Ahoy) over the store-brand (Chipsters) in a blind taste test. They tested this with 20 subjects and found that 14 (70%) chose Chips Ahoy as their favorite. They conducted a test of significance using simulation and got the following null distribution. (Note that this null distribution uses only 100 simulated samples and not the usual 1,000 or 10,000.) We will let π represent the long-run proportion of students that pick Chips Ahoy as their favorite. Using the null distribution above, what is the p-value when testing the hypotheses in part (a)? a. p-value = 59/100 = 0.59 b. p-value = 97/100 = 0.97 c. p-value = 3/100 = 0.03 d. p-value = 2/100 = 0.02

c/p-value = 3/100 = 0.03

Suppose 10 coins are flipped, and the proportion of heads is recorded. This process is repeated many, many times to develop a distribution of these sample proportions. What is the predicted mean and standard deviation for this distribution of sample proportions? Select one: A. Mean = 0.500, SD = 1.581 B. Mean = 5.000, SD = 1.581 C. Mean = 5.000, SD = 0.581 D. Mean = 0.500, SD = 0.158

d/Mean = 0.500, SD = 0.158