Exam 17: Inference About a Population Proportion

A December 2013 Gallup poll of 13,186 nationally representative adults found that 8.9% reported being sick with a cold. Based on these results, what is a 95% confidence interval for the proportion p of American adults who were sick with a cold that month?

A large fishing farm with thousands of fish has been treating its fish to stop a spreading fungal infection. A simple random sample of 50 fish is taken to determine the proportion p that is infected in this population. A careful examination determines that 6 of the fish sampled are infected. Using appropriate technology, what is a plus four 95% confidence interval for the population proportion of infected fish?

A large fishing farm with thousands of fish has been treating its fish to stop a spreading fungal infection. A simple random sample of 50 fish is taken to determine the proportion p that is infected in this population. A careful examination determines that 6 of the fish sampled are infected. The infection would be contained if fewer than 10% of the fish were infected. To answer this question, which null and alternative hypotheses would we test?

In 2013, Gallup surveyed a nationally representative sample of 2027 adults about their soda consumption. The survey showed that 24% mostly drank diet soda. This survey also found that 56% drank mostly either regular or diet soda. Is this evidence that the majority of American adults drink mostly either regular or diet soda? To answer this question, you test the following hypotheses: H₀: p = 0.50, H_a: p > 0.50 What is the z test statistic for this test?

The normal approximation to the sampling distribution of p̂ is most accurate when p is close to 0 or 1.

The American Veterinary Medical Association conducted a survey of veterinary clinics to estimate the proportion that do not treat large animals (e.g., cows, horses). The survey was mailed to a simple random sample of 120 veterinary clinics throughout the United States, and, 88 responded they do not treat large animals. What is the standard error SE^p̂ of the proportion p̂ of clinics that do not treat large animals?

People can develop allergic reactions to sulfites, a family of chemical compounds often used for commercial food preservation. Researchers examined a sample of 21 patients with hypertension who had sought treatment regarding some allergic reaction. The published findings state that 9 of them were sensitive to sulfites. What is a 95% plus four confidence interval for the true proportion of sulfite sensitivity among hypertensive patients with an allergy? Assume the necessary assumptions for this confidence interval are met.

Smaller margins of error call for _____________ sample sizes.

A survey of food allergies in a simple random sample of 184 young children found that 13 had some kind of food allergy. Why should we use the plus four method to compute a 95% confidence interval for the population proportion of young children who have a food allergy?

The American Veterinary Medical Association conducted a survey of veterinary clinics to estimate the proportion that do not treat large animals (e.g., cows, horses). The survey was mailed to a simple random sample of 120 veterinary clinics throughout the United States, and, 88 responded they do not treat large animals. What is a 95% confidence interval for p, the proportion of veterinary clinics that do treat large animals?

An inspector inspects a shipment of medications to determine the efficacy in terms of the proportion p in the shipment that failed to retain full potency after 60 days of production. Unless there is clear evidence that this proportion is less than 0.05, she will reject the shipment. To reach a decision, she will test the following hypotheses using the large-sample test for a population proportion: H₀: p = 0.05, H_a: p < 0.05 To do so, she selects a simple random sample of 200 pills. Suppose that 8 of the pills have failed to retain their full potency. Using appropriate technology, what is the P-value of her test?

A study examined the effectiveness of Botox injections in the corrugator supercilii muscles for the treatment of chronic migraines. Of the 29 migraine-prone patients participating in the study, 24 reported some form of improvement. Why can we not conclude from the findings that Botox injections in the corrugator supercilii muscles are the cause for these improvements?

A large fishing farm with thousands of fish has been treating its fish to stop a spreading fungal infection. A simple random sample of 50 fish is taken to determine the proportion p that is infected in this population. A careful examination determines that 6 of the fish sampled are infected. Using appropriate technology, obtain the P-value for the appropriate test. What should you conclude using a significance level of 0.05?

We want to assess possible side effects of the drug nifedipine, which is used to reduce chest pain in patients with angina. A study measures the heart rate in beats per minute of patients with angina before and after being treated with nifedipine. To estimate the proportion of subjects whose heart rate increases after taking nifedipine, which procedure should we use?

A TV news program conducts a call-in poll about a proposed city ban on smoking in public places. Of the 2467 callers, 1900 were opposed to the ban. Which of the following assumptions for inference about a proportion using a confidence interval is violated?

A phycologist is interested in determining the proportion of algae samples from a local rivulet that belong to a particular phylum. A simple random sample of 50 alga is obtained and each alga is categorized as either being a cyanobacterium or not. The researcher found that 38 samples were, in fact, cyanobacteria. Without relying on any previous knowledge, the researcher wanted to estimate the proportion of algae that were cyanobacteria with a margin of error of at most 0.01 in a 99% confidence interval. How large a sample size would be required?

There is growing concern over a possible causal link between consumption of diet soda and obesity. In 2013, Gallup surveyed a nationally representative sample of 2027 adults about their soda consumption. In the sample, 921 adults described themselves as overweight, and 295 of them said that they mostly drank diet soda. Based on these results, what is a 95% confidence interval for the proportion p of overweight American adults who mostly drink diet soda?

The American Veterinary Medical Association conducted a survey of veterinary clinics to estimate the proportion that do not treat large animals (e.g., cows, horses). The survey was mailed to a simple random sample of 120 veterinary clinics throughout the United States, and, 88 responded they do not treat large animals. Using technology and the plus four method, what is a 95% confidence interval for the proportion of veterinary clinics that do treat large animals?

A simple random sample of 900 13- to 17-year-olds found that 411 had responded better to a new drug therapy for autism. Let p be the proportion of all teens in this age range who respond better. What is a 95% confidence interval for p based on these data?

An inspector inspects a shipment of medications to determine the efficacy in terms of the proportion p in the shipment that failed to retain full potency after 60 days of production. Unless there is clear evidence that this proportion is less than 0.05, she will reject the shipment. To reach a decision, she will test the following hypotheses using the large-sample test for a population proportion: H₀: p = 0.05, H_a: p < 0.05 To do so, she selects a simple random sample of 200 pills. Suppose that 8 of the pills have failed to retain their full potency. Which of the following assumptions for inference about a proportion using a hypothesis test are violated in this scenario?