Deck 14: Exercises

Question 1

A researcher wishes to determine if aerobic exercise improves mental performance immediately following the exercise. He plans to have high school students participate in 30 minutes of aerobic exercise and then take a standard test of their reasoning skills. Suppose the scores of high school students on this test of reasoning skills immediately after 30 minutes of aerobic exercise follow a Normal distribution, with mean μ and standard deviation σ= 4. Suppose also that, in the general population of all high school students, scores on the test of reasoning skills follow a Normal distribution, with mean 25 and standard deviation σ= 4. The researcher, therefore, decides to test the following hypotheses:
H₀: μ= 25, H_a: μ> 25
To do so, the researcher has 10,000 high school students perform 30 minutes of aerobic exercise and then, immediately following the exercise, take the test. The mean score for these students is x̄= 25.2 and the P-value is less than 0.0001. It is appropriate to conclude which of the following?

A)The researcher has conclusively proved that, for high school students, 30 minutes of aerobic exercise substantially improves mental performance.
B)The researcher has strong evidence that, for high school students, 30 minutes of aerobic exercise substantially improves mental performance.
C)The researcher has moderate evidence that, for high school students, 30 minutes of aerobic exercise substantially improves mental performance.
D)None of these options is correct.

Answer 1

D

Question 2

A researcher wishes to determine if aerobic exercise improves mental performance immediately following the exercise. He plans to have high school students participate in 30 minutes of aerobic exercise and then take a standard test of their reasoning skills. Suppose the scores of high school students on this test of reasoning skills immediately after 30 minutes of aerobic exercise follow a Normal distribution, with mean μ and standard deviation σ= 4. Suppose also that, in the general population of all high school students, scores on the test of reasoning skills follow a Normal distribution, with mean 25 and standard deviation σ= 4. The researcher, therefore, decides to test the following hypotheses:
H₀: μ= 25, H_a: μ> 25
To do so, the researcher has 10,000 high school students perform 30 minutes of aerobic exercise and then, immediately following the exercise, take the test. The mean score for these students is x̄= 25.2 and the P-value is less than 0.0001.
Suppose that two high school students decide to see if they get the same results as the researcher. They both do 30 minutes of aerobic exercise and, immediately following the exercise, take the test. The mean of their scores is = 25.2 seconds, the same as that of the researcher. It is appropriate to conclude which of the following?

A)They have reproduced the results of the researcher, and their P-value will be the same as that of the researcher.
B)They have reproduced the results of the researcher, but their P-value will be slightly smaller than that of the researcher.
C)They will reach the same statistical conclusion as the researcher, but their P-value will be a bit different than that of the researcher.
D)None of these options is correct.

Answer 2

D

Question 3

The z procedures and many other procedures designed for Normal distributions rely on Normality of the sampling distribution of x̄.

Answer 3

True

Question 4

The z procedures and many other procedures designed for Normal distributions rely Normality of individual observations in the population.

Question 5

You plan to construct a confidence interval for the mean μ of a Normal population with (known) standard deviation σ. Which of the following actions will reduce the size of the margin of error?

A)Use a lower level of confidence.
B)Increase the sample size.
C)Reduce σ.
D)All of these options are correct.

Question 6

The distribution of total body protein in healthy adult men is approximately Normal, with standard deviation σ= 0.1 kg. A 95% confidence interval for the true mean total body protein of adult men with liver cirrhosis, based on 58 randomly selected adult males with liver cirrhosis, has a margin of error of 0.026 kg. How would a 90% confidence interval based on the same study compare to the 95% confidence interval?

A)The 90% confidence interval would have the same margin of error.
B)The 90% confidence interval would have a larger margin of error.
C)The 90% confidence interval would have a smaller margin of error.
D)The 90% confidence interval may have a different margin of error, but that would depend on the actual sample mean.

Question 7

The distribution of total body protein in healthy adult men is approximately Normal, with standard deviation σ= 0.1 kg. A 95% confidence interval for the true mean total body protein of adult men with liver cirrhosis, based on 58 randomly selected adult males with liver cirrhosis, has a margin of error of 0.026 kg. If the study had enrolled fewer subjects, how would the resulting margin of error for a 95% confidence interval compare?

A)It would be the same.
B)It would be larger.
C)It would be smaller.
D)It may be different, but that would depend on the actual sample mean.

Question 8

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels more than 200 mg/dl is Normally distributed, with mean (in milligrams) μ and standard deviation σ= 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels more than 200 mg/dl, but you believe that the mean decrease in cholesterol is actually less than advertised. To explore this question, you test the following hypotheses at the 0.05 level:
H₀: μ= 10, H_a: μ< 10
You obtain a P-value of 0.052. Which of the following is TRUE?

A)At the α= 0.05 significance level, you have proved that H₀ is true.
B)You have failed to obtain any evidence for H_a.
C)There is some evidence against H₀, and a study using a larger sample size may be worthwhile.
D)This should be viewed as a pilot study, and the data suggest that further investigation of the hypotheses will not be fruitful at the α= 0.05 significance level.

Question 9

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels more than 200 mg/dl is Normally distributed, with mean (in milligrams) μ and standard deviation σ= 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels more than 200 mg/dl, but you believe that the mean decrease in cholesterol is actually less than advertised. To explore this question, you test the following hypotheses at the 0.05 level:
H₀: μ= 10, H_a: μ< 10
You carry out the level 0.05 test by randomly selecting 100 people with cholesterol levels more than 200 mg/dl and, after they have eaten this brand of oatmeal for one month, computing the mean decrease x̄ in the subjects' cholesterol levels. What is the probability of a Type II error for your test at μ= 8.5?

A)0.0004
B)0.0012
C)0.9988
D)0.9996

Question 10

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels more than 200 mg/dl is Normally distributed, with mean (in milligrams, mg) μ and standard deviation σ= 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels more than 200 mg/dl, but you believe that the mean decrease in cholesterol is less than advertised. To explore this question, you test the following hypotheses:
H₀: μ= 10, H_a: μ< 10
You carry out the test at a significance level of 0.05 by randomly selecting 100 people with cholesterol levels more than 200 mg/dl and, after they have eaten this brand of oatmeal for one month, computing the mean decrease x̄ in the subjects' cholesterol levels. What is the probability of a Type I error for your test at μ= 8.5?

A)0.0012
B)0.0500
C)0.9500
D)0.9988

Question 11

A radio show runs a phone-in survey each morning. One morning the show host asked listeners whether the city should change regulations for new structures at local beachfront property to save certain sea grass populations. Most of those phoning in their responses answered, "No, it is just grass-there is no reason to worry about it," and the station reported the results as statistically significant. What may we safely conclude?

A)There is strong evidence that the city council should change the regulations.
B)It is unlikely that, if all Americans were asked their opinion, the result would differ from that obtained in the poll.
C)There is strong evidence that the city council should not change the regulations.
D)We can conclude very little other than that most of those phoning in their responses believe there is no reason to care about sea grass.

Question 12

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. A 95% confidence interval for the true mean total body protein of adult men with liver cirrhosis, based on 67 randomly selected adult males with liver cirrhosis, has a margin of error of 0.024 kg. How would a 99% confidence interval based on the same study compare to the 95% confidence interval?

A)A 99% confidence interval would have the same margin of error.
B)A 99% confidence interval would have a larger margin of error.
C)A 99% confidence interval would have a smaller margin of error.
D)A 99% confidence interval may have a different margin of error, but that would depend on the actual sample mean.

Question 13

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. A 95% confidence interval for the true mean total body protein of adult men with liver cirrhosis, based on 67 randomly selected adult males with liver cirrhosis, has a margin of error of 0.024 kg. If the study had enrolled more subjects, how would the resulting margin of error for a 95% confidence interval compare?

A)It would be the same.
B)It would be larger.
C)It would be smaller.
D)It may be different, but that would depend on the actual sample mean.

Question 14

Does 30 minutes of meditation every day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who meditated for 30 minutes each day for a period of 6 months. At the end of the study, 300 variables related to the mental performance of the subjects were measured on each subject, and the means compared to known means for these variables in the population of all adults. Sixteen of these variables were significantly better (in the sense of statistical significance) at the α= 0.05 level for the group who performed 30 minutes of meditation each day as compared to the population as a whole, and 3 variables were significantly better at the α= 0.01 level for the group who performed 30 minutes of meditation each day as compared to the population as a whole. What would be the correct conclusion?

A)There is very good statistical evidence that 30 minutes of meditation each day provides some improvement in mental performance.
B)There is very good statistical evidence that 30 minutes of meditation each day provides some improvement for the variable that was significant at the α= 0.01 level. We should be somewhat cautious about making claims for the variables that were significant at the α= 0.05 level.
C)These results would have provided very good statistical evidence that 30 minutes of meditation each day provides some improvement in mental performance if the number of subjects had been larger. It is premature to draw statistical conclusions from studies in which the number of subjects is smaller than the number of variables measured.
D)None of these options is correct.

Question 15

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. Which sample size, n, would be required to obtain a margin of error of 0.01 kg for a 95% confidence interval for the mean total body protein of adult men with liver cirrhosis?

A)20
B)200
C)385
D)400

Question 16

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. Which sample size, n, would be required to obtain a margin of error of 0.02 kg for a 90% confidence interval for the mean total body protein of adult men with liver cirrhosis?

A)9
B)68
C)83
D)96

Question 17

Which of the following describes good practice when assessing the validity of any test of hypotheses?

A)Examine the probability model that serves as a basis for the test by using exploratory data analysis on the datA)
B)Test the hypotheses at several different levels of significance.
C)Test both one- and two-sided hypotheses to help guarantee consistency.
D)All of these options are correct.

Question 18

Which of the following describe good practice when assessing the validity of any test of hypotheses?

A)Examine the probability model that serves as a basis for the test by using exploratory data analysis on the datA)
B)Determine exactly how the study was conducted.
C)Determine which assumptions the researchers made.
D)All of these options are correct.

Question 19

A researcher proposes a study to investigate a quantitative variable for which the population standard deviation is known to be 4. She wants a margin of error of 2.5 with 95% confidence. Which sample size should she use?

A)3
B)4
C)9
D)10

Question 20

You carry out a study to investigate the decline in cod populations off the coast of Canada. To carry out a hypothesis test, you must decide which sample size should be used. Which factors affect your decision?

A)
$$\alpha$$
B)
$$\beta$$
C)Effect size
D)All of these options are correct.

Question 21

The time (in number of days) until maturity of a certain variety of tomato plant is Normally distributed, with mean and standard deviation ?= 2.4. You select a simple random sample of four plants of this variety and measure the time until maturity. The four times, in days, are as follows:
63 $\quad$ 69 $\quad$ 62 $\quad$ 66 Based on these data, what is the 99% confidence interval for ?, in days?

A)65.00 ± 1.55
B)65.00 ± 2.35
C)65.00 ± 3.09
D)65.00 ± 4.07

Question 22

The time (in number of days) until maturity of a certain variety of tomato plant is Normally distributed, with mean and standard deviation ?= 2.4. You select a simple random sample of four plants of this variety and measure the time until maturity. The four times, in days, are as follows:
63 $\quad$ 69 $\quad$ 62 $\quad$ 66 If you wanted the margin of error for the 99% confidence interval to be ±1 inch, which size of simple random sample should you select?

A)2
B)7
C)16
D)39

Question 23

Your advisor has asked you to reduce the margin of error on your confidence interval from ±5 mg/l to ±1 mg/l. The population standard deviation is known to be 8 mg/. What should you do?

A)Increase the sample size to decrease the margin of error.
B)Decrease the confidence level, C, so that the margin of error will cover a narrower range.
C)Neither A nor B, because you need to decrease the sample size and increase the confidence level.
D)Both A and B, because both will help decrease the margin of error.

Question 24

The level of nitrogen oxides (NO_x) in the exhaust of cars of a particular model varies Normally, with standard deviation σ= 0.05 gram per mile (g/mi). Government regulations call for NO_x emissions no higher than 0.3 g/mi. In testing H₀: μ= 0.3 versus H_a: μ< 0.3 using significance level 0.05, we find that the power of this test against an alternative of μ= 0.28 based on a random sample of size n = 40 is 0.808. What impact would using a larger sample size (n = 50) have?

A)It would increase the power of this particular test.
B)It would decrease the power of this particular test.
C)It would increase the probability of a Type I error.
D)It would decrease the probability of a Type I error.

Question 25

People tend to give more socially acceptable answers when asked a sensitive question in person rather than anonymously via computer. When computing a 95% confidence interval for a population parameter, what does the margin of error cover?

A)It covers all deviations from the parameter that are due to answer bias.
B)It covers 95% of deviations from the parameter that are due to answer bias.
C)It covers deviations from the parameter that are due to answer bias as long as the sample was a random sample.
D)It does not cover the deviations from the parameter that are due to answer bias.

Question 26

A researcher plans to conduct a test of hypotheses at the α= 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value of the parameter of interest. What is the probability that the researcher will commit a Type I error?

A)0.05
B)0.20
C)0.80
D)Equal to the P-value and cannot be determined until the data have been collected

Question 27

A researcher plans to conduct a test of hypotheses at the α= 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value for the parameter of interest. What is the probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power?

A)0.05
B)0.20
C)0.80
D)Equal to 1 - (P-value) and cannot be determined until the data have been collected

Question 28

What is the consequence of rejecting the null hypothesis when, in fact, it is true?

A)We have committed a Type I error.
B)We have committed a Type II error.
C)We have computed a probability of being correct that is equal to the P-value.
D)We have computed a probability of being correct that is equal to 1 - (P-value).

Question 29

When performing a test of hypotheses, what is a Type II error?

A)Rejecting the null hypothesis when it is true
B)Railing to reject the null hypothesis when it is false
C)Incorrectly specifying the null hypothesis
D)Incorrectly specifying the alternative hypothesis

Question 30

Which of the following will increase the value of the power in a statistical test of hypotheses?

A)Increase the Type II error probability.
B)Increase the sample size.
C)Reject the null hypothesis only if the P-value is smaller than the level of significance.
D)All of the above options are correct.

Question 31

Which of the following is NOT an assumption included in the simple conditions for inference about a mean?

A)Simple random sampling with no nonresponse or other practical difficulty
B)Response variable with a perfect Normal distribution in the population
C)A population mean that is known
D)A population standard deviation that is known

Question 32

You collect a random sample of size n from a population; from the data collected, you compute a 95% confidence interval for the mean of the population. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data?

A)Use a larger confidence level.
B)Use a smaller confidence level.
C)Use the same confidence level but compute the interval n times. Approximately 5% of these intervals will be larger.
D)Nothing can guarantee absolutely that you will get a larger interval.We can say only that the chance of obtaining a larger interval is 0.05.

Question 33

A study randomly assigned 200 lab rats to 20 different groups: 2 control groups and 18 treatment groups in which the rats were fed various amounts of genetically modified corn exposed to the pesticide Roundup. The researchers ran a separate hypothesis test for each group and found a significant (P-value < 0.05) number of tumors for 2 treatment groups. The published findings received some criticism. Here is a quote about this study: "Critics point out that the type of rats used in this study are particularly prone to tumors. So if you divide up 200 of them into 20 groups, you are likely to get very high (statistically significant) tumor rates in some of the groups." What does this quote refer to?

A)The possibility that the statistically significant results are a Type I error
B)The possibility that the statistically significant results are a Type II error
C)The certainty that the statistically significant results are a Type I error
D)The certainty that the statistically significant results are a Type II error

Question 34

A study randomly assigned 200 lab rats to 20 different groups: 2 control groups and 18 treatment groups in which the rats were fed various amounts of genetically modified corn exposed to the pesticide Roundup. The researchers ran a separate hypothesis test for each group and found a significant (P-value < 0.05) number of tumors for 2 treatment groups. The published findings received some criticism. Here is a quote about this study: "[T]he fact that such clusters of tumors didn't show up in the two small control groups could easily be due to random chance." What does this quote refer to?

A)The possibility that the lack of statistical significance in the control groups is a Type I error
B)The possibility that the lack of statistical significance in the control groups is a Type II error
C)The certainty that the lack of statistical significance in the control groups is a Type I error
D)The certainty that the lack of statistical significance in the control groups is a Type II error

Question 35

A study examined the link between sodium intake and blood pressure in children and adolescents between the ages of 8 and 18. The mean systolic blood pressure (SBP) was 112.8 mm Hg among the 636 study participants who were overweight or obese and consumed high levels of sodium, resulting in a 95% confidence interval for this group of 110.7 to 114.9 mm Hg. The numerical value of the margin of error for this confidence interval is __________ mm Hg.

Question 36

A study examined the link between sodium intake and blood pressure in children and adolescents between the ages of 8 and 18. The mean systolic blood pressure (SBP) was 112.8 mm Hg among the 636 study participants who were overweight or obese and consumed high levels of sodium, resulting in a 95% confidence interval for this group of 110.7 to 114.9 mm Hg. If there had been more than 636 participants in this group, how would the resulting margin of error for a 95% confidence interval compare?

A)It would have been smaller than the original margin of error.
B)It would have been the same as the original margin of error.
C)It would have been larger than the original margin of error.
D)It may have been different than the original margin of error, but that would depend on the actual sample mean.

Question 37

A study examined the link between sodium intake and blood pressure in children and adolescents between the ages of 8 and 18. The mean systolic blood pressure (SBP) was 112.8 mm Hg among the 636 study participants who were overweight or obese and consumed high levels of sodium, resulting in a 95% confidence interval for this group of 110.7 to 114.9 mm Hg. How would the margin of error for a 99% confidence interval based on the same 636 study participants compare?

A)It would have a smaller margin of error than the 95% confidence interval.
B)It would have the same margin of error than the 95% confidence interval.
C)It would have a larger margin of error than the 95% confidence interval.
D)It would likely change, but we cannot tell if it would have a smaller or a larger margin of error.

Question 38

The makers of the pomegranate juice POM Wonderful claimed in advertisements that the juice helps with erectile dysfunction; this claim was based on a study showing a significant improvement after drinking POM Wonderful. However, the Federal Trade Commission (FTC) filed a formal complaint "alleging that the erectile dysfunction claims are false and unsubstantiated because the study on which the company relied did not show that POM Juice was any more effective than a placebo." What can we say about the FTC complaint?

A)It reflects the fact that statistical significance is not the same as practical significance.
B)It reflects the fact that studies must be replicated before any statistical claim can be made.
C)It shows that the study had not enrolled enough subjects to be convincing.
D)It shows that the FTC does not understand the concept of statistical significance.

Question 39

The pH measurements of water specimens from various locations along a given river basin are Normally distributed, with standard deviation σ= 0.3. The average pH of water specimens from 4 randomly selected locations on this river basin is x̄= 7.8. Based on these data, what is the value for the margin of error for a 95% confidence interval for μ?

A)0.094 kg
B)0.294 kg
C)0.427 kg
D)0.588 kg

Question 40

The pH measurements of water specimens from various locations along a given river basin are Normally distributed, with standard deviation σ= 0.3. The average pH of water specimens from 4 randomly selected locations on this river basin is x̄= 7.8. A 95% confidence interval for μ was computed for these data. If, instead of 4 water specimens, we had measured the pH of 16 water specimens, how would the margin of error of a 95% confidence interval for μ compare to the original 95% confidence interval based on a sample of 4 specimens?

A)It would be four times as large.
B)It would be twice as large.
C)It would be half as large.
D)It would be one-fourth as large.

Question 41

What does statistical significance depend on?

A)The size of the observed effect
B)The variability of individuals in the population
C)The sample size
D)All of these options are correct.

Question 42

Because large random samples have small chance variation, very small deviations from the null hypothesis can be highly significant if the sample is large.

Question 43

Because small random samples have a lot of chance variation, even large deviations from the null hypothesis can fail to be statistically significant if the sample is small.

Question 44

The P-value gives the probability that the data are the result of chance variations alone.