Deck 17: Inference About a Population Proportion

Question 1

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. What is the proportion p̂ of algae that were cyanobacteria?

A)0.24
B)0.38
C)0.50
D)0.76

Answer 1

D

Question 2

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. What is the standard error SE^p̂ of the proportion p̂ of the algae that were cyanobacteria?

A)0.0604
B)0.0036
C)0.1824
D)0.5000

Answer 2

A

Question 3

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?

A)13,572
B)13,573
C)16,512
D)16,513

Answer 3

D

Question 4

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)0.053 to 0.243
B)0.030 to 0.210
C)0.046 to 0.194
D)0.069 to 0.228

Question 5

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?

A)H₀:p = 0.50, H_a:p < 0.50
B)H₀:p = 0.12, H_a:p < 0.12
C)H₀:p = 0.12, H_a:p < 0.10
D)H₀:p = 0.10, H_a:p < 0.10

Question 6

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. Using appropriate technology, what is the numerical value of the test statistic for the correct set of hypotheses?

A)-5.374
B)0.00
C)0.471
D)0.681

Question 7

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?

A)You can conclude that the infection is contained.
B)You can conclude that the infection is not contained.
C)You should fail to reach a conclusion as to whether the infection is contained.
D)You can conclude that the population proportion of infected fish is 10% of the total population.

Question 8

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?

A)The counts of successes and failures in the sample are too small for the large-sample method.
B)The sample size is too small for the large-sample method.
C)We do not know sigma, the population standard deviation.
D)We do not know the population proportion of young children who have a food allergy.

Question 9

A survey of food allergies in a simple random sample of 184 young children found that 13 had some kind of food allergy. What is the plus four sample estimate of the proportion of young children who have a food allergy for this sample?

A)0.0707
B)0.0798
C)0.0815
D)0.0904

Question 10

A survey of food allergies in a simple random sample of 184 young children found that 13 had some kind of food allergy. What is the margin of error for a plus four 95% confidence interval for the true population proportion of young children who have a food allergy?

A)0.0311
B)0.0325
C)0.0370
D)0.0387

Question 11

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)The data are an SRS from the population of interest.
B)The population is at least 10 times as large as the sample.
C. n is so large that both the count of successes, np̂ and the count of failures, n(1-p̂), are 10 or greater.
D)There appear to be no violations of the assumptions.

Question 12

In 2013, Gallup surveyed a nationally representative sample of 2027 adults about their soda consumption. The survey showed that 24% mostly drank diet soda. Based on the results, what is a 95% confidence interval for the proportion of American adults who mostly drink diet soda?

A)0.24 ± 0.009
B)0.24 ± 0.019
C)0.24 ± 0.120
D)0.24 ± 0.182

Question 13

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?

A)5.40
B)3.30
C)1.12
D)0.56

Question 14

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 a total of 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 P-value for the test?

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

Question 15

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?

A)0.46 ± 0.017
B)0.46 ± 0.027
C)0.46 ± 0.033
D)0.46 ± 0.248

Question 16

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. Suppose you wanted to see whether the majority of teens in this age range respond better. To answer this question, you test the following hypotheses:
H₀: p = 0.50, H_a: p > 0.50
Using appropriate technology, what is the P-value of your 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 17

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)0.2352
B)0.2582
C)0.4704
D)0.5164

Question 18

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?

A)The data are an SRS from the population of interest.
B)The population is at least 10 times as large as the sample.
C)n is so large that both np₀ and n(1 - p₀) are 10 or more, where p₀ is the proportion with major defects if the null hypothesis is true.
D)There appear to be no violations.

Question 19

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 a 95% plus four confidence interval for the true proportion of pills in the shipment that have failed to retain full potency based on these data?

A)0.049 ± 0.0269
B)0.049 ± 0.0272
C)0.049 ± 0.0296
D)0.049 ± 0.0299

Question 20

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?

A)0.130 to 0.160
B)0.142 to 0.174
C)0.290 to 0.350
D)0.305 to 0.335

Question 21

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?

A)0.02
B)0.03
C)0.04
D)0.05

Question 22

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)(0.163, 0.371)
B)(0.188, 0.346)
C)(0.200, 0.353)
D)(0.629, 0.837)

Question 23

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?

A)(0.163, 0.371)
B)(0.188, 0.346)
C)(0.200, 0.353)
D)(0.629, 0.837)

Question 24

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. What is a plus four 95% confidence interval for the population proportion of migraine patients who benefit from such Botox injections? Assume the necessary assumptions for this confidence interval are met.

A)0.690 to 0.965
B)0.688 to 0.967
C)0.717 to 0.859
D)0.648 to 0.927

Question 25

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)There is no comparison group, and the improvements could simply be due to the placebo effect.
B)We computed only a confidence interval, not a P-value.
C)We are only 95% confident that our interval captures the true population parameter.
D)The sample size is too small.

Question 26

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.

A)0.44 ± 0.195
B)0.44 ± 0.214
C)0.44 ± 0.220
D)0.44 ± 0.245

Question 27

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. Why should we use the plus four method to compute a 95% confidence interval for the population proportion of sulfite sensitivity among patients with hypertension and an allergy?

A)We need the additional four observations to overcome bias.
B)The counts of successes and failures in the sample are too small for the large-sample method.
C)We do not know sigma, the population standard deviation.
D)We do not know the value of the population proportion.

Question 28

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. How large a sample n would you need to estimate p with margin of error 0.05 with 95% confidence? Use the guess p = 0.50 as the value for p*.

A)n = 157
B)n = 385
C)n = 769
D)n = 1537

Question 29

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, 16 reported complete migraine elimination. Using technology and a plus four 90% confidence interval for the population proportion of patients with complete migraine remission following such Botox injections, what is the margin of error? Assume the necessary assumptions for this confidence interval are valid.

A)0.0867
B)0.0923
C)0.1426
D)0.1810

Question 30

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, 16 reported complete migraine elimination. How large a sample n would you need to estimate p with margin of error 0.02 with 90% confidence? Use the guess p* = 0.50 as the value for p.

A)n = 846
B)n = 1678
C)n = 1692
D)n = 2401

Question 31

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

Question 32

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)0.089 ± 0.005
B)0.089 ± 0.010
C)0.089 ± 0.015
D)0.089 ± 0.030

Question 33

A December 2013 Gallup poll of 13,186 nationally representative adults found that 8.9% reported being sick with a cold. In its report, Gallup warned that, "In addition to sampling error, question wording and practical difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls." Which of the following sources of uncertainty is taken into account when computing the margin of error of the confidence interval?

A)Question wording
B)Response bias
C)Nonresponse
D)Sampling error

Question 34

A study examined a simple random sample of smokers who tried to quit smoking by using either a nicotine patch or e-cigarettes. Here is a quote from a news article referencing the study: "Quitting is hard. The number of people who quit (within 6 months) was low overall-just 38 of the 584 smokers given the e-cigarette or the patch." Using technology, what is the plus four sample estimate of the proportion of smokers who quit within 6 months using either e-cigarettes or the nicotine patch?

A)0.0451
B)0.0650
C)0.0680
D)0.0851

Question 35

A study examined a simple random sample of smokers who tried to quit smoking by using either a nicotine patch or e-cigarettes. Here is a quote from a news article referencing the study: "Quitting is hard. The number of people who quit (within 6 months) was low overall-just 38 of the 584 smokers given the e-cigarette or the patch." What is the margin of error for a plus four 95% confidence interval for the true population proportion of smokers who quit within 6 months using either e-cigarettes or the nicotine patch?

A)0.0884
B)0.0477
C)0.0407
D)0.0204

Question 36

A study examined a simple random sample of smokers who tried to quit smoking by using either a nicotine patch or e-cigarettes. Here is a quote from a news article referencing the study: "Quitting is hard. The number of people who quit (within 6 months) was low overall-just 38 of the 584 smokers given the e-cigarette or the patch." Are the conditions for this inference procedure met?

A)Yes, all the conditions are met.
B)No, there are too few successes.
C)No, the study was not double-blind.
D)No, we do not know the value of the population proportion.

Question 37

Smaller margins of error call for _____________ sample sizes.

A)smaller
B)larger
C)equal

Question 38

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

Question 39

Before performing a hypothesis test for a proportion H₀: p = p₀, what must we demonstrate?

A) $n p _ { 0 } \geq 10$
B) $n \left( 1 - p _ { 0 } \right) \geq 10$
C) $n \geq 30$
D)Both a and b