Exam 18: Comparing Two Proportions

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 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 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?

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 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 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 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?

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?

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?

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?

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 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 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?

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 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?

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?

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?

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 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?

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 ______________.