Deck 16: Comparing Two Means

Question 1

A study randomly assigned 60 adults diagnosed with metabolic syndrome to two groups. One group was given daily tablets containing the Chinese herbal supplement Yiqi Huaju Qingli, while the other group was given daily placebo tablets. All 60 patients maintained their standard medical treatment throughout the 12 weeks of the study. Urinary microalbumin (in mg/l) was assessed for all participants at the beginning and at the end of the study. Urinary microalbumin is a key factor in metabolic syndrome and an early sign of diabetic nephropathy. The researchers wanted to see if, by the end of the study, urinary microalbumin was significantly lower in the herbal supplement group than in the placebo group. To answer this question, which hypothesis test should the researchers use?

A)A one-sample t test
B)A matched pairs t test
C)A two-sample t test
D)Any of the above are valid; it just needs to be a t test since σis unknown.

Answer 1

C

Question 2

A study randomly assigned 60 adults diagnosed with metabolic syndrome to two groups. One group was given daily tablets containing the Chinese herbal supplement Yiqi Huaju Qingli, while the other group was given daily placebo tablets. All 60 patients maintained their standard medical treatment throughout the 12 weeks of the study. Urinary microalbumin (in mg/l) was assessed for all participants at the beginning and at the end of the study. Urinary microalbumin is a key factor in metabolic syndrome and an early sign of diabetic nephropathy. The researchers also wanted to see if urinary microalbumin was significantly lower after 12 weeks in the herbal supplement group. To answer this question, which hypothesis test should the researchers use?

A)A one-sample t test
B)A matched pairs t test
C)A two-sample t test
D)Any of the above are valid; it just needs to be a t test since σis unknown.

Answer 2

B

Question 3

When are the two-Sample t procedures accurate?

A)When both populations are Normal and both sample sizes n₁ and n₂ are 5 or larger
B)When at least one of the populations is Normal and both sample sizes n₁ and n₂ are 5 or larger
C)When both populations are Normal and only one of sample sizes n₁ and n₂ is 5 or larger
D)When both populations are Normal and the combined sample size n₁ + n₂ is 5 or larger

Answer 3

A

Question 4

A large P-value implies that H₀ is necessarily true,

Question 5

Researchers are interested in whether the effects of DDT poisoning in rats are less extreme in males. An SRS of 100 male rats and an independent SRS of 100 female rats are obtained. The male rats had a sample mean of 39 and a sample standard deviation of 8. The female rats had a sample mean of 40 and a standard deviation of 10. Let μ₁ represent the true mean response for the males and μ₂ represent the true mean response of the females. Which hypothesis should be tested?

A)H₀: μ₁ = μ₂ versus H_a: μ₁ < μ₂
B)H_0: μ_{Difference 1-2} = 0 versus H_a: μ_{Difference 1-2} ≠ 0
C)H₀: μ₁ = 39 and μ₂= 40 versus H_a: μ₁ < 39 and μ₂< 40
D)None of the above

Question 6

Researchers are interested in whether the effects of DDT poisoning in rats are less extreme in males. An SRS of 100 male rats and an independent SRS of 100 female rats are obtained. The male rats had a sample mean of 39 and a sample standard deviation of 8. The female rats had a sample mean of 40 and a standard deviation of 10. Let μ₁ represent the true mean response for the males and μ₂ represent the true mean response of the females. They test the following hypotheses:
H₀: μ₁ = μ₂ versus H_a: μ₁ < μ₂
The numerical value of the t statistic = .078 and the P-value = 0.22. What should we conclude?

A)Reject the null hypothesis at significance level alpha 5%
B)Fail to reject the null hypothesis at significance level alpha 5%
C)Reject the alternative hypothesis at significance level alpha 5%
D)Fail to reject the alternative hypothesis at significance level alpha 5%

Question 7

Researchers are interested in whether the effects of DDT poisoning in rats are less extreme in males. An SRS of 100 male rats and an independent SRS of 100 female rats are obtained. The male rats had a sample mean of 39 and a sample standard deviation of 8. The female rats had a sample mean of 40 and a standard deviation of 10. Let μ₁ represent the true mean response for the males and μ₂ represent the true mean response of the females. A 95% confidence interval for the difference in true mean response μ_{1 -} μ₂ is found to be ( _ 3.53, 1.53). What can we conclude?

A)Reject the null hypothesis at significance level alpha 5%
B)Fail to reject the null hypothesis at significance level alpha 5%
C)Reject the alternative hypothesis at significance level alpha 5%
D)Fail to reject the alternative hypothesis at significance level alpha 5%

Question 8

Researchers examined the bill color (in hue degree) of male and female zebra finches. Here are some summary statistics from the study:
$$\begin{array} { | l | l | l | l | } \hline \text { Sex } & \boldsymbol { n } & \text { Mean } & \text { Standard Deviation } \\\hline \text { Males } & 59 & 2.91 & 1.46 \\\hline \text { Females } & 60 & 7.42 & 2.48 \\\hline\end{array}$$ We wish to address the question, How different are male and female zebra finches in bill color? What is the margin of error for a 95% confidence interval for the difference in mean bill color ?_F-?_M (in hue degree)?

A)1.40
B)1.02
C)0.83
D)0.74

Question 9

Researchers examined the bill color (in hue degree) of male and female zebra finches. Here are some summary statistics from the study:
$$\begin{array} { | l | l | l | l | } \hline \text { Sex } & \boldsymbol { n } & \text { Mean } & \text { Standard Error } \\\hline \text { Males } & 59 & 2.91 & 0.19 \\\hline \text { Females } & 60 & 7.42 & 0.32 \\\hline\end{array}$$ We wish to address the question, How different are male and female zebra finches in bill color? What is a 95% confidence interval for the difference in mean bill color ?_F-?_M (in hue degree)?

A)4.41 to 4.61
B)3.77 to 5.25
C)3.23 to 5.79
D)1.20 to 10.22

Question 10

Researchers examined the bill color (in hue degree) of male and female zebra finches. Here are the summary statistics from the study:
$$\begin{array} { | l | l | l | l | } \hline \text { Sex } & \boldsymbol { n } & \text { Mean } & \text { Standard Error } \\\hline \text { Males } & 59 & 2.91 & 0.19 \\\hline \text { Females } & 60 & 7.42 & 0.32 \\\hline\end{array}$$ We are interested in testing the following hypotheses:
H₀: ?_M=?_F versus ?_M??_F
What can we conclude from the 95% confidence interval for the difference in mean bill color ?_F-?_M (in hue degree)?

A)Reject the null hypothesis at significance level alpha 5%
B)Fail to reject the null hypothesis at significance level alpha 5%
C)Reject the alternative hypothesis at significance level alpha 5%
D)Fail to reject the alternative hypothesis at significance level alpha 5%

Question 11

Researchers measured the amount of plasma carotenoids (in micrograms per milliliter) of male and female zebra finches. Here are the summary statistics from the study:
$$\begin{array} { | l | l | l | l | } \hline \text { Sex } & \boldsymbol { n } & \text { Mean } & \text { Standard Error } \\\hline \text { Males } & 50 & 31.77 & 1.85 \\\hline \text { Females } & 52 & 29.74 & 2.02 \\\hline\end{array}$$ Do male and female zebra finches differ in their plasma carotenoid levels? To answer this question, the researchers tested the following hypotheses:
?
H₀: ?_M=?_F versus ?_M??_F
What is the numerical value of the two-sample t statistic?

A)0.46
B)0.74
C)3.41
D)5.30

Question 12

Researchers measured the amount of plasma carotenoids (in micrograms per milliliter) of male and female zebra finches. Here are the summary statistics from the study:
$$\begin{array} { | l | l | l | l | } \hline \text { Sex } & \boldsymbol { n } & \text { Mean } & \text { Standard Error } \\\hline \text { Males } & 50 & 31.77 & 1.85 \\\hline \text { Females } & 52 & 29.74 & 2.02 \\\hline\end{array}$$ Do male and female zebra finches differ in their plasma carotenoid levels? To answer this question, the researchers tested the following hypotheses:
?
H₀: ?_M=?_F versus ?_M??_F
What is the P-value for the test?

A)Greater than 0.10
B)Between 0.10 and 0.05
C)Between 0.05 and 0.01
D)Less than 0.01

Question 13

Researchers measured the amount of plasma carotenoids (in micrograms per milliliter) of male and female zebra finches. Here are the summary statistics from the study:
$$\begin{array} { | l | l | l | l | } \hline \text { Sex } & \boldsymbol { n } & \text { Mean } & \text { Standard Error } \\\hline \text { Males } & 50 & 31.77 & 1.85 \\\hline \text { Females } & 52 & 29.74 & 2.02 \\\hline\end{array}$$ Do male and female zebra finches differ in their plasma carotenoid levels? To answer this question, the researchers tested the following hypotheses:
?
H₀: ?_M=?_F versus ?_M??_F
What can we conclude at significance level 5%?

A)Reject the alternative hypothesis
B)Fail to reject the alternative hypothesis
C)Reject the null hypothesis
D)Fail to reject the null hypothesis

Question 14

What is the rationale for avoiding the pooled procedures for inference?

A)Testing for the equality of variances is an unreliable procedure that is not robust.
B)The two-sample t, or "unequal variances procedure," is valid regardless of whether the two variances are actually unequal.
C)The two-sample t, or "unequal variances procedure," is almost always more accurate than the pooled procedure.
D)All of these options are correct.

Question 15

A new diet has been developed for raising beef cows. Two random samples of size 9 are independently selected, and one is given the standard diet while the other is given the new diet. After 18 weeks, the weight gain was measured for each cow. It was found that x̄₁= 30 with S₁= 8 and x̄₂= 26 with S₂= 6. Let μ₁ and μ₂ represent the mean weight gains we would observe for the entire population of beef cows when on, respectively, a new diet and a standard diet. Assume that two-sample t procedures are safe to use. Suppose the researcher had wished to test the following hypotheses:
H₀: μ₁ = μ₂ versus H_a: μ₁ > μ₂
What is the numerical value of the two-sample t statistic for this test?

A)0.36
B)1.20
C)2.57
D)4.00

Question 16

A new diet has been developed for raising beef cows. Two random samples of size 9 are independently selected, and one is given the standard diet while the other is given the new diet. After 18 weeks, the weight gain was measured for each cow. It was found that x̄₁= 30 with S₁= 8 and x̄₂= 26 with S₂= 6. Let μ₁ and μ₂ represent the mean weight gains we would observe for the entire population of beef cows when on, respectively, a new diet and a standard diet. Assume that two-sample t procedures are safe to use. Suppose the researcher had wished to test the following hypotheses:
H₀: μ₁ = μ₂ versus H_a: μ₁ > μ₂
What is the P-value for the test of these hypotheses?

A)Greater than 0.10
B)Between 0.10 and 0.05
C)Between 0.05 and 0.01
D)Less than 0.01

Question 17

A new diet has been developed for raising beef cows. Two random samples of size 9 are independently selected, and one is given the standard diet while the other is given the new diet. After 18 weeks, the weight gain was measured for each cow. It was found that x̄₁= 30 with S₁= 8 and x̄₂= 26 with S₂= 6. Let μ₁ and μ₂ represent the mean weight gains we would observe for the entire population of beef cows when on, respectively, a new diet and a standard diet. Assume that two-sample t procedures are safe to use. About which of the following characteristics would it have been most important that the researchers be blind during the experiment?

A)Whether the cows were male or female
B)The purpose of the research
C)The initial weight of the cows
D)Which diet the cows received

Question 18

A new diet has been developed for raising beef cows. Two random samples of size 9 are independently selected, and one is given the standard diet while the other is given the new diet. After 18 weeks, the weight gain was measured for each cow. It was found that x̄₁= 30 with S₁= 8 and x̄₂= 26 with S₂= 6. Let μ₁ and μ₂ represent the mean weight gains we would observe for the entire population of beef cows when on, respectively, a new diet and a standard diet. Assume that two-sample t procedures are safe to use. To which value are the degrees of freedom for these t procedures closest?

A)14
B)12
C)9
D)8

Question 19

The circumference of a species of tree was thought to be different depending on certain geographical factors. An SRS of 60 trees of this species is selected in Sequoyah National Forest, where it is found that x̄₁= 6 meters with a standard deviation S₁= 3 meters. A second SRS of size 40 trees of the same species is independently selected in Yellowstone National Forest, and it is found that x̄₂= 4 meters with a standard deviation S₂= 2 meters. Let μ₁ and μ₂ represent, respectively, the true mean circumferences of this species of tree. Assume the two-sample t procedures are safe to use. What is a 95% confidence interval for μ₁ - μ₂ based on these data?

A)2 ± 0.50 m
B)2 ± 0.84 m
C)2 ± 0.99 m
D)2 ± 1.34 m

Question 20

The circumference of a species of tree was thought to be different depending on certain geographical factors. An SRS of 60 trees of this species is selected in Sequoyah National Forest, where it is found that x̄₁= 6 meters with a standard deviation S₁= 3 meters. A second SRS of size 40 trees of the same species is independently selected in Yellowstone National Forest, and it is found that x̄₂= 4 meters with a standard deviation S₂= 2 meters. Let μ₁ and μ₂ represent, respectively, the true mean circumferences of this species of tree. Assume the two-sample t procedures are safe to use. The researcher tests the following hypotheses:
H₀: μ₁ = μ₂ versus H_a: μ₁ ≠ μ₂
What is the numerical value of the two-sample t statistic testing these hypotheses?

A)2.00
B)3.00
C)4.00
D)8.00

Question 21

The circumference of a species of tree was thought to be different depending on certain geographical factors. An SRS of 60 trees of this species is selected in Sequoyah National Forest, where it is found that x̄₁= 6 meters with a standard deviation S₁= 3 meters. A second SRS of size 40 trees of the same species is independently selected in Yellowstone National Forest, and it is found that x̄₂= 4 meters with a standard deviation S₂= 2 meters. Let μ₁ and μ₂ represent, respectively, the true mean circumferences of this species of tree. Assume the two-sample t procedures are safe to use. The researcher tests the following hypotheses:
H₀: μ₁ = μ₂ versus H_a: μ₁ ≠ μ₂
What is the P-value for the test of these hypotheses?

A)Greater than 0.10
B)Between 0.10 and 0.05
C)Between 0.05 and 0.01
D)Less than 0.01

Question 22

The circumference of a species of tree was thought to be different depending on certain geographical factors. An SRS of 60 trees of this species is selected in Sequoyah National Forest, where it is found that x̄₁= 6 meters with a standard deviation S₁= 3 meters. A second SRS of size 40 trees of the same species is independently selected in Yellowstone National Forest, and it is found that x̄₂= 4 meters with a standard deviation S₂= 2 meters. Let μ₁ and μ₂ represent, respectively, the true mean circumferences of this species of tree. Assume the two-sample t procedures are safe to use. To which value are the degrees of freedom for these t procedures closest?

A)99
B)98
C)60
D)50

Question 23

The circumference of a species of tree was thought to be different depending on certain geographical factors. An SRS of 60 trees of this species is selected in Sequoyah National Forest, where it is found that x̄₁= 6 meters with a standard deviation S₁= 3 meters. A second SRS of size 40 trees of the same species is independently selected in Yellowstone National Forest, and it is found that x̄₂= 4 meters with a standard deviation S₂= 2 meters. Let μ₁ and μ₂ represent, respectively, the true mean circumferences of this species of tree. Assume the two-sample t procedures are safe to use. The researcher tests the following hypotheses:
H₀: μ₁ = μ₂ versus H_a: μ₁ ≠ μ₂
The 90% confidence interval is 2 ± 0.83 meters. What conclusion can we state based on this confidence interval?

A)We would reject the null hypothesis of no difference at the 0.10 level.
B)The P-value is less than 0.10.
C)Neither A nor B is correct.
D)Both A and B are correct.

Question 24

The circumference of a species of tree was thought to be different depending on certain geographical factors. An SRS of 60 trees of this species is selected in Sequoyah National Forest, where it is found that x̄₁= 6 meters with a standard deviation S₁= 3 meters. A second SRS of size 40 trees of the same species is independently selected in Yellowstone National Forest, and it is found that x̄₂= 4 meters with a standard deviation S₂= 2 meters. Let μ₁ and μ₂ represent, respectively, the true mean circumferences of this species of tree. Assume the two-sample t procedures are safe to use. The researcher tests the following hypotheses:
H₀: μ₁ = μ₂ versus H_a: μ₁ ≠ μ₂
Which of the following statements is the correct interpretation for 99% confidence interval?

A)There is a 95% chance that the difference in population means is between 0.69 and
3.31 meters.
B)There is a 95% chance that a randomly selected individual from this population would have a
Response between 0.69 and 3.31 meters.
C)We are 95% confident that the difference in sample means is between 0.69 and 3.31
Meters.
D)We are 95% confident that the difference in population means is between 0.69 and 3.31 meters.

Question 25

A study examined the effect of protein consumption on weight gain in rats. Investigators obtained 14 female rats and randomly assigned 7 of them to a low-protein diet and the other 7 to a high-protein diet. After 60 days, the weight gained by each rat was recorded (in grams). Here are the summary findings, displayed as "mean (standard deviation)":
$$\begin{array}{l}\begin{array} { l l } \text { Low-Protein Diet}&\text { High-Protein Diet}\\101.3(20.6)&127.5(21.1)\end{array}\end{array}$$ Researchers want to know if the amount of protein in the rats' diet significantly affects their weight gain. What are the corresponding null and alternative hypotheses?

A)H₀: ?_High = ?_Low versus H_a: ?_High > ?_Low
B)H₀: ?_High = ?_Low versus H_a: ?_High ? ?_Low
C)H₀: ?_High 127.5 and ?_low = 101.3 versus H_a: ?_High > ?_Low
D)H₀: ?_High = ?_Low versus H_a: ?_High 127.5 and ?_Low = 101.3

Question 26

A study examined the effect of protein consumption on weight gain in rats. Investigators obtained 14 female rats and randomly assigned 7 of them to a low-protein diet and the other 7 in a high-protein diet. After 60 days, the weight gained by each rat was recorded (in grams). Here are the summary findings, displayed as "mean (standard deviation)":
$$\begin{array}{l}\begin{array} { l l } \text { Low-Protein Diet}&\text { High-Protein Diet}\\101.3(20.6)&127.5(21.1)\end{array}\end{array}$$ Researchers want to know if the amount of protein in the rats' diet significantly affects their weight gain. Based on the data, what is the absolute numerical value of t statistic for the appropriate test?

A)1.25
B)2.35
C)6.04
D)13.10

Question 27

A study examined the effect of protein consumption on weight gain in rats. Investigators obtained 14 female rats and randomly assigned 7 of them to a low-protein diet and the other 7 in a high-protein diet. After 60 days, the weight gained by each rat was recorded (in grams). Here are the summary findings, displayed as "mean (standard deviation)":
$$\begin{array}{l}\begin{array} { l l } \text { Low-Protein Diet}&\text { High-Protein Diet}\\101.3(20.6)&127.5(21.1)\end{array}\end{array}$$ Researchers want to know if the amount of protein in the rats' diet significantly affects their weight gain.
What should they conclude at significance level 5%?

A)Reject the alternative hypothesis
B)Fail to reject the alternative hypothesis
C)Reject the null hypothesis
D)Fail to reject the null hypothesis

Question 28

A study examined the effect of protein consumption on weight gain in rats. Investigators obtained 14 female rats and randomly assigned 7 of them to a low-protein diet and the other 7 in a high-protein diet. After 60 days, the weight gained by each rat was recorded (in grams). Here are the summary findings, displayed as "mean (standard deviation)":
$$\begin{array}{l}\begin{array} { l l } \text { Low-Protein Diet}&\text { High-Protein Diet}\\101.3(20.6)&127.5(21.1)\end{array}\end{array}$$ Given that the degrees of freedom for the appropriate t procedure is approximately 12, what is a 95% confidence interval for ?_High - ?_Low?

A)1.9 to 50.5 grams
B)15.1 to 37.3 grams
C)20.6 to 21.1 grams
D)101.3 to 127.5 grams

Question 29

A study examined the effect of protein consumption on weight gain in rats. Investigators obtained 14 female rats and randomly assigned 7 of them to a low-protein diet and the other 7 in a high-protein diet. After 60 days, the weight gained by each rat was recorded (in grams). Here are the summary findings, displayed as "mean (standard deviation)":
$$\begin{array}{l}\begin{array} { l l } \text { Low-Protein Diet}&\text { High-Protein Diet}\\101.3(20.6)&127.5(21.1)\end{array}\end{array}$$ Which of the following statements would lead us to believe that the t procedures were NOT safe to use here?

A)The experiment was not double-blinded.
B)The distributions of the data were skewed or had outliers.
C)The population standard deviations were not known.
D)The t procedures are always safe to use in a randomized experiment.

Question 30

Some chronic conditions produce episodes of severe flare-ups ("active" stage) between healthy periods ("inactive" stage). The active stage often necessitates immediate medical treatment. Researchers looked for biomarkers that might help identify these active stages. Here are data on the relative abundance of the protein BPG0235 in two independent random samples of men with the same chronic condition, 7 men in the active phase and 9 men in the inactive phase:
$$\begin{array} { | l | l | } \hline \text { Active } & \text { Inactive } \\\hline 1.0 & 1.1 \\\hline 1.0 & 0.9 \\\hline 1.0 & 0.5 \\\hline 1.2 & 0.9 \\\hline 1.2 & 1.0 \\\hline 1.3 & 1.0 \\\hline 1.0 & 0.8 \\\hline & 0.7 \\\hline & 1.1 \\\hline\end{array}$$ We want to know if there is a significant difference in the relative abundance of the protein BPG0235 between men in the active stage and men in the inactive phase. Therefore, we test the following hypotheses:
?
H₀: ?_A= ?₁ versus H_a: ?_A? ?₁
What is the absolute numerical value of the two-sample t statistic?

A)13.7
B)0.22
C)3.41
D)2.58

Question 31

Some chronic conditions produce episodes of severe flare-ups ("active" stage) between healthy periods ("inactive" stage). The active stage often necessitates immediate medical treatment. Researchers looked for biomarkers that might help identify these active stages. Here are data on the relative abundance of the protein BPG0235 in two independent random samples of men with the same chronic condition, 7 men in the active phase and 9 men in the inactive phase:
$$\begin{array} { | l | l | } \hline \text { Active } & \text { Inactive } \\\hline 1.0 & 1.1 \\\hline 1.0 & 0.9 \\\hline 1.0 & 0.5 \\\hline 1.2 & 0.9 \\\hline 1.2 & 1.0 \\\hline 1.3 & 1.0 \\\hline 1.0 & 0.8 \\\hline & 0.7 \\\hline & 1.1 \\\hline\end{array}$$ We want to know if there is a significant difference in the relative abundance of the protein BPG0235 between men in the active stage and men in the inactive phase. Therefore, we test the following hypotheses:
?
H₀: ?_A= ?₁ versus H_a: ?_A? ?₁
What is the closest value to the P-value for the test?

A)9%
B)2%
C)1%
D)0.01%

Question 32

Some chronic conditions produce episodes of severe flare-ups ("active" stage) between healthy periods ("inactive" stage). The active stage often necessitates immediate medical treatment. Researchers looked for biomarkers that might help identify these active stages. Here are data on the relative abundance of the protein BPG0235 in two independent random samples of men with the same chronic condition, 7 men in the active phase and 9 men in the inactive phase:
$$\begin{array} { | l | l | } \hline \text { Active } & \text { Inactive } \\\hline 1.0 & 1.1 \\\hline 1.0 & 0.9 \\\hline 1.0 & 0.5 \\\hline 1.2 & 0.9 \\\hline 1.2 & 1.0 \\\hline 1.3 & 1.0 \\\hline 1.0 & 0.8 \\\hline & 0.7 \\\hline & 1.1 \\\hline\end{array}$$ We want to know if there is a significant difference in the relative abundance of the protein BPG0235 between men in the active stage and men in the inactive phase. Therefore, we test the following hypotheses:
?
H₀: ?_A= ?₁ versus H_a: ?_A? ?₁
Which assumption must be true for the test results to be correct?

A)The population distributions must be roughly Normal.
B)The sample standard deviations must be similar.
C)The population standard deviations must be equal.
D)The same individuals must be measured during an active phase and during an inactive phase.

Question 33

Some chronic conditions produce episodes of severe flare-ups ("active" stage) between healthy periods ("inactive" stage). The active stage often necessitates immediate medical treatment. Researchers looked for biomarkers that might help identify these active stages. Here are data on the relative abundance of the protein BPG0199 in two independent random samples of men with the same chronic condition, 7 men in the active phase and 9 men in the inactive phase:
$$\begin{array} { | l | l | } \hline \text { Active } & \text { Inactive } \\\hline 2.6 & 1.2 \\\hline 2.7 & 2.1 \\\hline 2.0 & 1.8 \\\hline 2.4 & 1.5 \\\hline 1.9 & 1.9 \\\hline 1.4 & 3.2 \\\hline 1.8 & 1.3 \\\hline & 1.9 \\\hline & 1.6 \\\hline\end{array}$$ We want to know if there is a significant difference in the relative abundance of the protein BPG0199 between men in the active stage and men in the inactive phase. Therefore, we test the following hypotheses:
?
H₀: ?_A= ?₁ versus H_a: ?_A? ?₁
The test degrees of freedom is roughly 13. Which value is the P-value for the test closest to?

A)0.001
B)0.03
C)0.1
D)0.3

Question 34

Some chronic conditions produce episodes of severe flare-ups ("active" stage) between healthy periods ("inactive" stage). The active stage often necessitates immediate medical treatment. Researchers looked for biomarkers that might help identify these active stages. Here are data on the relative abundance of the protein BPG0199 in two independent random samples of men with the same chronic condition, 7 men in the active phase and 9 men in the inactive phase:
$$\begin{array} { | l | l | } \hline \text { Active } & \text { Inactive } \\\hline 2.6 & 1.2 \\\hline 2.7 & 2.1 \\\hline 2.0 & 1.8 \\\hline 2.4 & 1.5 \\\hline 1.9 & 1.9 \\\hline 1.4 & 3.2 \\\hline 1.8 & 1.3 \\\hline & 1.9 \\\hline & 1.6 \\\hline\end{array}$$ We want to know if there is a significant difference in the relative abundance of the protein BPG0199 between men in the active stage and men in the inactive phase. Therefore, we test the following hypotheses:
?
H₀: ?_A= ?₁ versus H_a: ?_A? ?₁
Which assumption must be TRUE for this test result to be correct?

A)The same individuals must be measured during an active phase and during an inactive phase.
B)The population standard deviations must be equal.
C)The sample standard deviations must be similar.
D)The population distributions must be roughly Normal.

Question 35

Which of the following procedures is NOT robust to non-Normality?

A)The one-sample t test
B)The t test for matched pairs
C)The two-sample t test
D)The F test for comparing two population standard deviations

Question 36

A study randomly assigned 60 adults diagnosed with metabolic syndrome to two groups, each containing 30 patients. One group was given daily tablets containing the Chinese herbal supplement Yiqi Huaju Qingli, while the other group was given daily placebo tablets. All 60 patients maintained their standard medical treatment throughout the 12 weeks of the study. Urinary microalbumin is a key factor in metabolic syndrome and an early sign of diabetic nephropathy. Urinary microalbumin (in mg/l) was assessed for all participants at the beginning and at the end of the study. Here are the published findings, expressed as "mean (standard deviation)":
$$\begin{array} { | l | l | l | } \hline & \text { Herbal Supplement } & \text { Placebo } \\\hline \text { Start of study } & 101.58 ( 75.39 ) & 98.50 ( 66.92 ) \\\hline \text { End of study } & 47.11 ( 27.85 ) & 78.87 ( 60.65 ) \\\hline\end{array}$$ Is there evidence that urinary microalbumin differed significantly between the two groups at the start of the study? Based on the data, what is the absolute numerical value of the t statistic for the appropriate test?

A)0.17
B)0.87
C)1.54
D)3.08

Question 37

A study randomly assigned 60 adults diagnosed with metabolic syndrome to two groups, each containing 30 patients. One group was given daily tablets containing the Chinese herbal supplement Yiqi Huaju Qingli, while the other group was given daily placebo tablets. All 60 patients maintained their standard medical treatment throughout the 12 weeks of the study. Urinary microalbumin is a key factor in metabolic syndrome and an early sign of diabetic nephropathy. Urinary microalbumin (in mg/l) was assessed for all participants at the beginning and at the end of the study. Here are the published findings, expressed as "mean (standard deviation)":
?
$$\begin{array} { | l | l | l | } \hline & \text { Herbal Supplement } & \text { Placebo } \\\hline \text { Start of study } & 101.58 ( 75.39 ) & 98.50 ( 66.92 ) \\\hline \text { End of study } & 47.11 ( 27.85 ) & 78.87 ( 60.65 ) \\\hline\end{array}$$ Using significance level ?= 0.05, do the data provide evidence that urinary microalbumin differed significantly between the two groups at the start of the study? [Note that the degrees of freedom is roughly 40 for this test.]

A)Yes, P < 0.05
B)Yes, P > 0.05
C)No, P < 0.05
D)No, P > 0.05

Question 38

A study randomly assigned 60 adults diagnosed with metabolic syndrome to two groups, each containing 30 patients. One group was given daily tablets containing the Chinese herbal supplement Yiqi Huaju Qingli, while the other group was given daily placebo tablets. All 60 patients maintained their standard medical treatment throughout the 12 weeks of the study. Urinary microalbumin is a key factor in metabolic syndrome and an early sign of diabetic nephropathy. Urinary microalbumin (in mg/l) was assessed for all participants at the beginning and at the end of the study. Here are the published findings, expressed as "mean (standard deviation)":
$$\begin{array} { | l | l | l | } \hline & \text { Herbal Supplement } & \text { Placebo } \\\hline \text { Start of study } & 101.58 ( 75.39 ) & 98.50 ( 66.92 ) \\\hline \text { End of study } & 47.11 ( 27.85 ) & 78.87 ( 60.65 ) \\\hline\end{array}$$ Is there evidence that urinary microalbumin is significantly lower in the herbal group than in the placebo group at the end of the study? Based on the data, what is the absolute numerical value of the t statistic for the appropriate test?

A)0.53
B)1.06
C)2.61
D)3.71

Question 39

A study randomly assigned 60 adults diagnosed with metabolic syndrome to two groups, each containing 30 patients. One group was given daily tablets containing the Chinese herbal supplement Yiqi Huaju Qingli, while the other group was given daily placebo tablets. All 60 patients maintained their standard medical treatment throughout the 12 weeks of the study. Urinary microalbumin is a key factor in metabolic syndrome and an early sign of diabetic nephropathy. Urinary microalbumin (in mg/l) was assessed for all participants at the beginning and at the end of the study. Here are the published findings, expressed as "mean (standard deviation)":
?
$$\begin{array} { | l | l | l | } \hline & \text { Herbal Supplement } & \text { Placebo } \\\hline \text { Start of study } & 101.58 ( 75.39 ) & 98.50 ( 66.92 ) \\\hline \text { End of study } & 47.11 ( 27.85 ) & 78.87 ( 60.65 ) \\\hline\end{array}$$ Using significance level ?= 0.05, do the data provide evidence that urinary microalbumin is significantly lower in the herbal group than in the placebo group at the end of the study? [Note that the degrees of freedom is roughly 40 for this test.]

A)Yes, P < 0.05
B)Yes, P > 0.05
C)No, P < 0.05
D)No, P > 0.05

Question 40

A study randomly assigned 60 adults diagnosed with metabolic syndrome to two groups, each containing 30 patients. One group was given daily tablets containing the Chinese herbal supplement Yiqi Huaju Qingli, while the other group was given daily placebo tablets. All 60 patients maintained their standard medical treatment throughout the 12 weeks of the study. Urinary microalbumin is a key factor in metabolic syndrome and an early sign of diabetic nephropathy. Urinary microalbumin (in mg/l) was assessed for all participants at the beginning and at the end of the study. Here are the published findings, expressed as "mean (standard deviation)":
?
$$\begin{array} { | l | l | l | } \hline & \text { Herbal Supplement } & \text { Placebo } \\\hline \text { Start of study } & 101.58 ( 75.39 ) & 98.50 ( 66.92 ) \\\hline \text { End of study } & 47.11 ( 27.85 ) & 78.87 ( 60.65 ) \\\hline\end{array}$$ Given that the degrees of freedom for the appropriate t procedure is roughly 40, what is a 95% confidence interval for ?_Herbal¯?_Placebo at the end of the study?

A)-31.76 ± 12.31 mg/l
B)-31.76 ± 16.40 mg/l
C)-31.76 ± 24.61 mg/l
D)-31.76 ± 49.22 mg/l