Deck 18: Comparing Two Proportions

Question 1

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. What is the estimate for the proportion of adults who use prescription drugs and who are concerned?

A)0.183
B)0.525
C)0.537
D)0.825

Answer 1

D

Question 2

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. What is the estimate for the proportion of adults who do not use prescription drugs and who are concerned?

A)0.183
B)0.340
C)0.525
D)0.537

Answer 2

C

Question 3

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. What is the standard error of the sampling distribution for the difference in the sample proportions,p̂₁ -p̂₂ ?

A)0.005
B)0.023
C)0.152
D)0.300

Answer 3

B

Question 4

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. Let p₁ and p₂ be the proportion of all prescription drug users and nonusers, respectively, who are concerned regarding health care. Using technology, what is a 90% confidence interval for p₁-p₂?

A)0.300 ± 0.023
B)0.300 ± 0.038
C)0.300 ± 0.045
D)0.300 ± 0.054

Question 5

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. Consider prescription drug users to be the reference group. What is the relative risk of being concerned?

A)0.6362
B)1.1034
C)1.5718
D)4.7000

Question 6

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. Consider prescription drug users to be the reference group. What is the odds ratio of being concerned?

A)0.2348
B)1.1034
C)4.2594
D)4.7000

Question 7

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. Consider prescription drug users to be the reference group. What is the relative risk reduction?

A)-0.5718
B)-0.3638
C)0.3638
D)0.5718

Question 8

A random digit dialing sample of 1750 adults found that 1140 used some form of prescription medication. Of those who used prescription drugs, 940 said they are concerned about the current state of health care; 320 of the 610 nonusers agreed with this position as well. Consider prescription drug users to be the reference group. What is the absolute risk reduction?

A)-0.3638
B)-0.3000
C)0.3000
D)0.3638

Question 9

A phytologist is worried that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a simple random sample of 50 algae from a local river and an independent simple random sample of 75 algae from the local coastline. He finds that 22 algae in the river sample and 66 algae in the coastline sample are cyanobacteria. He considers the river to act as a control group. Let p₁ and p₂ represent the proportion of cyanobacteria in riparian (i.e., river) and coastal populations, respectively. Using technology, what is a 99% confidence interval for p₁-p₂?

A)-0.44 ± 0.0163
B)-0.44 ± 0.1896
C)-0.44 ± 0.2054
D)-0.44 ± 0.2149

Question 10

A phytologist is worried that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a simple random sample of 50 algae from a local river and an independent simple random sample of 75 algae from the local coastline. He finds that 22 algae in the river sample and 66 algae in the coastline sample are cyanobacteria. He considers the river to act as a control group. Let p₁ and p₂ represent the proportion of cyanobacteria in riparian (i.e., river) and coastal populations, respectively. Is there evidence that the proportion of cyanobacteria is higher in coastal samples than in riparian samples? To determine this, you test the following hypotheses:
H₀:p₁ = p₂, H_a:p₁ < p₂
​
Using technology, what is the value of the z test statistic for testing these hypotheses?

A)z = -5.98
B)z = -5.78
C)z = -5.58
D)z = -5.28

Question 11

A phytologist is worried that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a simple random sample of 50 algae from a local river and an independent simple random sample of 75 algae from the local coastline. He finds that 22 algae in the river sample and 66 algae in the coastline sample are cyanobacteria. He considers the river to act as a control group. Let p₁ and p₂ represent the proportion of cyanobacteria in riparian (i.e., river) and coastal populations, respectively. Is there evidence that the proportion of cyanobacteria is higher in coastal samples than in riparian samples? To determine this, you test the following hypotheses:
H₀:p₁ = p₂, H_a:p₁ < p₂
​
Using technology, what is the P-value of your test?

A)Between 0.10 and 0.05
B)Between 0.05 and 0.01
C)Between 0.01 and 0.001
D)Less than 0.001

Question 12

A phytologist is worried that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a simple random sample of 50 algae from a local river and an independent simple random sample of 75 algae from the local coastline. He finds that 22 algae in the river sample and 66 algae in the coastline sample are cyanobacteria. He considers the river to act as a control group. Let p₁ and p₂ represent the proportion of cyanobacteria in riparian (i.e., river) and coastal populations, respectively. What is the relative risk of algae being cyanobacteria in coastal areas?

A)0.11
B)0.50
C)2.00
D)9.33

Question 13

A phytologist is worried that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a simple random sample of 50 algae from a local river and an independent simple random sample of 75 algae from the local coastline. He finds that 22 algae in the river sample and 66 algae in the coastline sample are cyanobacteria. He considers the river to act as a control group. Let p₁ and p₂ represent the proportion of cyanobacteria in riparian (i.e., river) and coastal populations, respectively. What is the odds ratio for coastal to riparian environments of algae being cyanobacteria?

A)0.1071
B)0.7857
C)7.3333
D)9.3333

Question 14

A phytologist is worried that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a simple random sample of 50 algae from a local river and an independent simple random sample of 75 algae from the local coastline. He finds that 22 algae in the river sample and 66 algae in the coastline sample are cyanobacteria. He considers the river to act as a control group. Let p₁ and p₂ represent the proportion of cyanobacteria in riparian (i.e., river) and coastal populations, respectively. What is the relative risk reduction from coastal to riparian environments of algae being cyanobacteria?

A)-1.0000
B)-0.5000
C)0.5000
D)1.0000

Question 15

A phytologist is worried that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a simple random sample of 50 algae from a local river and an independent simple random sample of 75 algae from the local coastline. He finds that 22 algae in the river sample and 66 algae in the coastline sample are cyanobacteria. He considers the river to act as a control group. Let p₁ and p₂ represent the proportion of cyanobacteria in riparian (i.e., river) and coastal populations, respectively. What is the absolute risk reduction from coastal to riparian environments of algae being cyanobacteria?

A)-0.4400
B)-0.1071
C)0.4400
D)0.1071

Question 16

Is there a difference in the proportion of individuals with food allergies between young children and adults? We plan to test the following hypotheses:
​
H₀:p_Children = p_Adults, H₀:p_Children ≠ p_Adults
​
A survey finds that 13 among a simple random sample of 184 young children have some food allergy, compared with 4 out of 163 adults. What is the pooled proportion of individuals with a food allergy based on these data?

A)0.0461
B)0.0476
C)0.0490
D)0.0500

Question 17

Is there a difference in the proportion of individuals with food allergies between young children and adults? We plan to test the following hypotheses:
​
H₀:p_Children = p_Adults, H₀:p_Children ≠ p_Adults
A survey finds that 13 among a simple random sample of 184 young children have some food allergy, compared with 4 out of 163 adults. Using technology, what is the test statistic for this hypothesis test?

A)0.046
B)1.986
C)2.055
D)2.851

Question 18

Is there a difference in the proportion of individuals with food allergies between young children and adults? We plan to test the following hypotheses:
​
H₀:p_Children = p_Adults, H₀:p_Children ≠ p_Adults
A survey finds that 13 among a simple random sample of 184 young children have some food allergy, compared with 4 out of 163 adults. Using technology, what is the P-value of this test?

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

Question 19

A survey finds that 13 among a simple random sample of 184 young children have some food allergy, compared with 4 out of 163 adults. We want to compute a 90% plus four confidence interval for P_Childre¯ P_Adults . What is the plus four estimate for the proportion of young adults who have a food allergy?

A)0.0707
B)0.0753
C)0.0798
D)0.0904

Question 20

A survey finds that 13 among a simple random sample of 184 young children have some food allergy, compared with 4 out of 163 adults. We want to compute a 90% plus four confidence interval for P_Childre¯ P_Adults. What is the plus four estimate for the proportion of adults who have a food allergy?

A)0.0245
B)0.0296
C)0.0303
D)0.0348

Question 21

A survey finds that 13 among a simple random sample of 184 young children have some food allergy, compared with 4 out of 163 adults. We want to compute a 90% plus four confidence interval for P_Childre¯ P_Adults. Using technology, what is a 90% plus four confidence interval for P_Childre¯ P_Adults?

A)0.045 ± 0.039
B)0.045 ± 0.046
C)0.045 ± 0.084
D)0.045 ± 1.645

Question 22

The Virginia Department of Environmental Quality (VDEQ) uses probabilistic monitoring to regulate the water quality of streams in the Commonwealth of Virginia. Of the 85 Eastern Virginia Sites (group 1), 17 do not meet minimum requirements. Of the 80 units sampled in Western Virginia Sites (group 2), 24 do not meet minimum requirements. Assume the data can be treated as independent simple random samples. What are the null and alternative hypotheses to test that the proportions of streams that fail to meet minimum requirements in the two areas are equal?

A) $H _ { 0 } : p _ { 1 } = p _ { 2 } , H _ { a } : p _ { 1 } < p _ { 2 }$
B) $H _ { 0 } : p _ { 1 } = p _ { 2 } , H _ { a } : p _ { 1 } > p _ { 2 }$
C) $H _ { 0 } : p _ { 1 } = p _ { 2 } , H _ { a } : p _ { 1 } \neq p _ { 2 }$
D) $H _ { 0 } : p _ { 1 } \neq p _ { 2 } , H _ { a } : p _ { 1 } = p _ { 2 }$

Question 23

The Virginia Department of Environmental Quality (VDEQ) uses probabilistic monitoring to regulate the water quality of streams in the Commonwealth of Virginia. Of the 85 Eastern Virginia Sites (group 1), 17 do not meet minimum requirements. Of the 80 units sampled in Western Virginia Sites (group 2), 24 do not meet minimum requirements. Assume the data can be treated as independent simple random samples. What is the value of the z test statistic for testing the equality of the proportions of streams that fail to meet minimum requirements in the two areas?

A)z = -1.49
B)z = -1.65
C)z = -1.96
D)z = -2.33

Question 24

The Virginia Department of Environmental Quality (VDEQ) uses probabilistic monitoring to regulate the water quality of streams in the Commonwealth of Virginia. Of the 85 Eastern Virginia Sites (group 1), 17 do not meet minimum requirements. Of the 80 units sampled in Western Virginia Sites (group 2), 24 do not meet minimum requirements. Assume the data can be treated as independent simple random samples. What is the P-value of the test for equality of the proportions of streams that fail to meet minimum requirements in the two areas?

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

Question 25

A study recorded the body mass index (BMI) of a simple random sample of 200 American men and 200 American women. Individuals with a BMI greater than 30 are considered obese. To estimate the difference in the percentage of individuals who are obese among American men and American women, which inference procedure should you use?

A)One-sample z procedure for a proportion
B)Two-sample z procedure for proportions
C)One-sample or matched pairs z or t procedure for a mean
D)Two-sample z or t procedure for means

Question 26

Avandia is an antidiabetic drug and, like all drugs, it can have side effects. One study of patients with congestive heart failure found that 5 of the 110 subjects randomly assigned to take Avandia died of a cardiovascular event during the duration of the study, compared with 4 of the 114 subjects assigned to the placebo. The research question is "Is there evidence that patients with congestive heart failure have a higher probability of dying of a cardiovascular event if they take Avandia than if they take a placebo?" What is the pooled proportion of subjects who died of a cardiovascular event during the study?

A)0.0402
B)0.0450
C)0.0482
D)0.0500

Question 27

Avandia is an antidiabetic drug and, like all drugs, it can have side effects. One study of patients with congestive heart failure found that 5 of the 110 subjects randomly assigned to take Avandia died of a cardiovascular event during the duration of the study, compared with 4 of the 114 subjects assigned to the placebo. The research question is "Is there evidence that patients with congestive heart failure have a higher probability of dying of a cardiovascular event if they take Avandia than if they take a placebo?" Using technology, what is the test statistic for this hypothesis test?

A)0.395
B)1.354
C)2.055
D)2.851

Question 28

Avandia is an antidiabetic drug and, like all drugs, it can have side effects. One study of patients with congestive heart failure found that 5 of the 110 subjects randomly assigned to take Avandia died of a cardiovascular event during the duration of the study, compared with 4 of the 114 subjects assigned to the placebo. The research question is "Is there evidence that patients with congestive heart failure have a higher probability of dying of a cardiovascular event if they take Avandia than if they take a placebo?" Using technology, what is the P-value of this test?

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

Question 29

Avandia is an antidiabetic drug and, like all drugs, it can have side effects. One study of patients with congestive heart failure found that 5 of the 110 subjects randomly assigned to take Avandia died of a cardiovascular event during the duration of the study, compared with 4 of the 114 subjects assigned to the placebo. The research question is "Is there evidence that patients with congestive heart failure have a higher probability of dying of a cardiovascular event if they take Avandia than if they take a placebo?" What is the relative risk of cardiovascular death when taking Avandia compared with taking a placebo?

A)1.30
B)1.25
C)0.77
D)0.23

Question 30

There is growing concern over a possible causal link between consumption of diet soda and obesity. In 2013, Gallup took a nationally representative sample of 2027 adults and asked them about their soda consumption. In the sample, 921 adults described themselves as overweight, and 295 of those individuals also said that they mostly drink diet soda. The remaining 1106 adults described themselves as about the right weight, and 210 of those individuals also said that they mostly drink diet soda. We plan to test the following hypotheses about diet soda consumption:
H₀:p_Overweight = p_{About right}, Ha:p_Overweight ≠ p_{About right}
​
What is the pooled proportion of individuals who mostly drink diet soda?

A)0.1899
B)0.2491
C)0.3203
D)0.5000

Question 31

There is growing concern over a possible causal link between consumption of diet soda and obesity. In 2013, Gallup took a nationally representative sample of 2027 adults and asked them about their soda consumption. In the sample, 921 adults described themselves as overweight, and 295 of those individuals also said that they mostly drink diet soda. The remaining 1106 adults described themselves as about the right weight, and 210 of those individuals also said that they mostly drink diet soda. We plan to test the following hypotheses about diet soda consumption:
​
H₀:p_Overweight = p_{About right}, Ha:p_Overweight ≠ p_{About right}
​
Using technology, what is the test statistic for this hypothesis test?

A)1.39
B)1.986
C)5.81
D)6.76

Question 32

There is growing concern over a possible causal link between consumption of diet soda and obesity. In 2013, Gallup took a nationally representative sample of 2027 adults and asked them about their soda consumption. In the sample, 921 adults described themselves as overweight, and 295 of those individuals also said that they mostly drink diet soda. The remaining 1106 adults described themselves as about the right weight, and 210 of those individuals also said that they mostly drink diet soda. We plan to test the following hypotheses about diet soda consumption:
​
H₀:p_Overweight = p_{About right}, Ha:p_Overweight ≠ p_{About right}
​
Using technology, what is the P-value of this test?

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

Question 33

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growth), whereas all 10 of the 10 mice in the control group developed metastases. We want to compare the proportions of mice developing metastases under the two conditions. Which of the following statements is correct regarding this hypothesis test?

A)The test is not appropriate because the respective counts of successes and failures are not large enough.
B)The test is not appropriate because the experiment is not double-blind.
C)The test is appropriate because a total of 20 mice is large enough for any inference.
D)The test is appropriate because the study was randomized.

Question 34

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growths), whereas all 10 of the 10 mice in the control group developed metastases. What is the plus four estimate for the proportion of mice developing metastases in the treatment group?

A)0.1000
B)0.1667
C)0.2143
D)0.5000

Question 35

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growths), whereas all 10 of the 10 mice in the control group developed metastases. What is the plus four estimate for the proportion of mice developing metastases in the control group?

A)0.5000
B)0.8571
C)0.9167
D)1.0000

Question 36

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growths), whereas all 10 of the 10 mice in the control group developed metastases. What is the margin of error for a 95% plus four confidence interval for $$p \text { Treatment } = p \text { Control }$$ ?

A)0.186
B)0.263
C)0.282
D)0.500

Question 37

A pioneer study randomly assigned pregnant women to either a daily multivitamin supplement (including folic acid) or a placebo. In the 2104 pregnancies in the women taking the vitamin regimen, there were 28 cases of congenital malformation. In the 2052 pregnancies in the women taking the placebo, 47 cases of congenital malformation were recorded. What is the relative risk reduction, RRR, in congenital malformation when taking a multivitamin supplement compared with taking a placebo?

A)0.58
B)0.42
C)0.03
D)0.01

Question 38

A pioneer study randomly assigned pregnant women to either a daily multivitamin supplement (including folic acid) or a placebo. In the 2104 pregnancies in the women taking the vitamin regimen, there were 28 cases of congenital malformation. In the 2052 pregnancies in the women taking the placebo, 47 cases of congenital malformation were recorded. What is the absolute risk reduction, ARR, in congenital malformation when taking a multivitamin supplement compared with taking a placebo?

A)0.58
B)0.42
C)0.03
D)0.01

Question 39

A pioneer study randomly assigned pregnant women to either a daily multivitamin supplement (including folic acid) or a placebo. In the 2104 pregnancies in the women taking the vitamin regimen, there were 28 cases of congenital malformation. In the 2052 pregnancies in the women taking the placebo, 47 cases of congenital malformation were recorded. Rounded to the nearest whole number, what is the number needed to treat, NNT, to prevent one congenital malformation when taking a multivitamin supplement compared with taking a placebo?

A)1053
B)104
C)100
D)58

Question 40

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growths), whereas all 10 of the 10 mice in the control group developed metastases. Call (1) the treatment population (injected with antibodies) and (2) the reference population (untreated controls).
The absolute risk reduction, ARR, in metastases development when treated with tumor-specific antigens compared with no treatment is ______________.

Question 41

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growths), whereas all 10 of the 10 mice in the control group developed metastases. Call (1) the treatment population (injected with antibodies) and (2) the reference population (untreated controls).
The relative risk, RR, in metastases development when treated with tumor-specific antigens compared with no treatment is ______________.

Question 42

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growths), whereas all 10 of the 10 mice in the control group developed metastases. Call (1) the treatment population (injected with antibodies) and (2) the reference population (untreated controls).
The relative risk reduction, RRR, in metastases development when treated with tumor-specific antigens compared with no treatment is ______________.

Question 43

Could cancer cells be selectively targeted by using antibodies recognizing a tumor-specific protein marker? Researchers grafted human cancerous cells onto 20 healthy adult mice and then randomly assigned 10 of these mice to be treated with tumor-specific antibodies. They found that only 1 of the 10 mice treated with antibodies developed metastases (i.e., new cancer growths), whereas all 10 of the 10 mice in the control group developed metastases. Call (1) the treatment population (injected with antibodies) and (2) the reference population (untreated controls).
What is the number needed to treat, NNT, to prevent one mouse from developing metastases when treated with tumor-specific antigens compared with no treatment?

A)1
B)1.11
C)9
D)10

Question 44

There is growing concern over a possible causal link between consumption of diet soda and obesity. Gallup took in 2013 a nationally representative sample of 2027 adults and asked them about their soda consumption. In the sample, 921 adults described themselves as overweight, and 32% of those individuals said that they mostly drink diet soda. The remaining 1106 adults described themselves as about the right weight, and 19% of those individuals said that they mostly drink diet soda. What is the odds ratio of drinking mostly diet soda, using overweight individuals as the target population and about-right-weight individuals as the reference population?

A)1.2
B)1.7
C)2.0
D)9.1

Question 45

We cannot draw a conclusion about causation from an observational study even if the P-value is statistically significant because of confounding variables.

Question 46

The large-sample confidence interval for p₁ - p₂ generally has a true confidence level ___________the level asked for.

A)greater than
B)less than
C)equal to

Question 47

When using the plus four confidence interval for the difference p₁ - p₂, what do we add?

A)Two successes to each of the two samples
B)One success and one failure to each of the two samples
C)Two failures to each of the two samples
D)Four outcomes to the smallest sample