Unit 6 Inference with Proportions

A hypothesis test was conducted to investigate whether the population proportion of students at a certain college who went to the movie theater last weekend is greater than 0.2. A random sample of 100 students at this college resulted in a test statistic of 2.25. Assuming all conditions for inference were met, which of the following is closest to the p-value of the test?

0.0122

Decrease Probability of Type I Error

Decrease significance level

Increase Probability of Type II Error

Decrease significance level

Educators are testing a new program designed to help children improve their reading skills. The null hypothesis of the test is that the program does not help children improve their reading skills. For the educators, the more consequential error would be that the program does not help children improve their reading skills but the test indicated that it does help. Which of the following should the researchers do to avoid the more consequential error?

Decrease the significance level to decrease the probability of Type I error.

Consider a situation in which sampling without replacement is used to generate a random sample from each of two separate populations. To calculate a confidence interval to estimate the difference between population proportions, which of the following checks must be made?

Each population must be at least 10 times as large as its corresponding sample. The 10 percent rule checks that the sample is small relative to the population (that is, that the population is at least 10 times as large as the sample size) so that independence of observations within a sample is reasonable to assume.

A one-sample z-test for a population proportion p will be conducted. Which of the following conditions checks that the sampling distribution of the sample proportion is approximately normal? I. The sample is selected at random. II. np0≥10 and n(1−p0)≥10 for sample size n. III. The sample size is less than or equal to 10 percent of the population size.

II only The random sample condition is a check for independence, not normality. Large Counts rule -> normality

Decrease Probability of Type II Error (Avoid)

Increase sample size

Increase Probability of Type I Error

Increase significance level

Medical researchers are testing a new surgical procedure designed to minimize the side effects of surgery. The null hypothesis is that the procedure is not effective in minimizing side effects. For the researchers, the more consequential error would be that the procedure actually is effective in minimizing the side effects, but the test does not detect the effectiveness of the procedure. Which of the following should the researchers do to avoid the more consequential error?

Increase the significance level to increase the probability of a Type I error.

Which of the following is not a condition for constructing a confidence interval to estimate the difference between two population proportions?

The data must come from populations with approximately normal distributions.

A newspaper article claims that 92 percent of teens use social media. To investigate the claim, a polling organization selected a random sample of 100 teens, and 96 teens in the sample indicated that they use social media. Given the data, why is it not appropriate to use a one-sample z-test for a proportion to test the newspaper's claim?

The expected number of teens in the sample who do not use social media is less than 10.

In a hypothesis test for a single proportion, which of the following is assumed for the calculation of the p-value?

The null hypothesis is true.

A social scientist believed that less than 30 percent of adults in the United States watch 15 or fewer hours of television per week. To test the belief, the scientist randomly selected 1,250 adults in the United States. The sample proportion of adults who watch 15 or fewer hours of television per week was 0.28, and the resulting hypothesis test had a p-value of 0.061. The computation of the p-value assumes which of the following is true?

The population proportion of adults who watch 15 or fewer hours of television per week is 0.30. Since the scientist believes the proportion to be less than 0.30, the alternative hypothesis is Ha:p<0.30Ha:p<0.30. The null hypothesis is that the population proportion is 0.30.

For a one-sample test for a population proportion p and sample size n, why is it necessary that np0 and n(1−p0) are both at least 10 ?

The sample size must be large enough to support an assumption that the sampling distribution of the sample proportion is approximately normal. The condition justifies the assumption that the sampling distribution of pˆp^ is approximately normal.

Which of the following gives the probability of making a Type I error?

The significance level

At a large company, employees can take a course to become certified to perform certain tasks. There is an exam at the end of the course that needs to be passed for certification. The current pass rate is 0.7, but a new program is being tested to help increase the pass rate. The null hypothesis of the test is that the pass rate for the new program is 0.7. The alternative is that the pass rate for the new program is greater than 0.7. Which of the following describes a Type II error that could result from the test?

The test does not provide convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.8. A Type II error occurs if a false null hypothesis is not rejected; in other words, the test fails to find evidence for a true alternative hypothesis. In this case, the program is effective in raising the pass rate (the rate is 0.8), but the test fails to detect it (not finding evidence that the program is greater than 0.7).

A 95 percent confidence interval for the proportion difference p1−p2 was calculated to be (−0.12,0.17). Which of the following conclusions is supported by the interval?

There is not sufficient evidence to determine which proportion is greater.

A state biologist is investigating whether the proportion of frogs in a certain area that are bullfrogs has increased in the past ten years. The proportion ten years ago was estimated to be 0.20. From a recent random sample of 150 frogs in the area, 36 are bullfrogs. The biologist will conduct a test of H0:p=0.20 versus Ha:p>0.20. Which of the following is the test statistic for the appropriate test?

z=0.24−0.20/√(0.20)(0.80)/150

A factory manager selected a random sample of parts produced on an old assembly line and a random sample of parts produced on a new assembly line. The difference between the sample proportion of defective parts made on the old assembly line and the sample proportion of defective parts made on the new assembly line (old minus new) was 0.006. Under the assumption that all conditions for inference were met, a hypothesis test was conducted with the alternative hypothesis being the proportion of defective parts made on the old assembly line is greater than that of the new assembly line. The p-value of the test was 0.018. Which of the following is the correct interpretation of the p-value?

If there is no difference in the proportions of all defective parts made on the two assembly lines, the probability of observing a difference of at least 0.006 is 0.018.

Is the significance level of a hypothesis test equivalent to the probability that the null hypothesis is true?

No, the significance level is the probability of rejecting the null hypothesis when the null hypothesis is actually true.

There are 1,000 golden delicious and 1,000 red delicious apples in a cooler. In a random sample of 75 of the golden delicious apples, 48 had blemishes. In a random sample of 75 of the red delicious apples, 42 had blemishes. Assume all conditions for inference have been met. Which of the following is closest to the interval estimate of the difference in the numbers of apples with blemishes (golden delicious minus red delicious) at a 98 percent level of confidence?

(−105,265) The 98 percent confidence interval for the difference in proportions of apples with blemishes is (−0.105,0.265)(−0.105,0.265). The interval estimate for the difference in the numbers of apples with blemishes is found by multiplying the endpoints of the interval for the proportion by 1,000.

A gardener wants to know if soaking seeds in water before planting them increases the proportion of seeds that germinate. To investigate, the gardener will assign 50 seeds to be soaked before planting and 50 seeds to be planted without being soaked. After two weeks, the gardener will record how many seeds in each group germinated and construct a 95 percent confidence interval for the difference in proportions. Which of the following conditions for inference should be met? The seeds should be randomly assigned to a treatment. The group sizes should be less than 10 percent of the population sizes. The number of successful seeds and unsuccessful seeds in each group should be at least 10.

I and III only Because this is an experiment, random assignment to the treatments is a necessary condition. Also, the sample size needs to be sufficiently large to support the normality of the sampling distribution.

Biologists were studying the proportions of cats that had spotted markings on their fur in two populations of cats, C and F. An independent random sample of cats was taken from each population, and the difference between the sample proportions of cats with the spotted markings (C minus F) was 0.62. Under the assumption that all conditions for inference were met, a hypothesis test was conducted with the alternative hypothesis being that the population proportions are not equal. The p-value of the test was 0.01. Which of the following is the correct interpretation of the p-value?

If the proportions of all cats with spotted markings is the same for both populations, the probability of observing a sample difference of at least 0.62 or at most −0.62−0.62 is 0.01.

Researchers are studying the distribution of subscribers to a certain streaming service in different populations. From a random sample of 200 people in City C, 34 were found to subscribe to the streaming service. From a random sample of 200 people in City K, 54 were found to subscribe to the streaming service. Assuming all conditions for inference are met, which of the following is a 90 percent confidence interval for the difference in population proportions (City C minus City K) who subscribe to the streaming service?

(0.17−0.27)±1.65√(0.17)(0.83)+(0.27)(0.73)/200

Researchers investigating a new drug selected a random sample of 200 people who are taking the drug. Of those selected, 76 indicated they were experiencing side effects from the drug. If 5,000 people took the drug, which of the following is closest to the interval estimate of the number of people who would indicate they were experiencing side effects from the drug at a 90 percent level of confidence?

(1620,2180) The 90 percent confidence interval for the proportion of people who would indicate they were experiencing side effects from the drug is (0.324,0.436). The interval estimate for the number of people who would indicate they were experiencing side effects from the drug is found by multiplying the endpoints of the interval for the proportions by 5,000.

A major credit card company is interested in the proportion of individuals who use a competitor's credit card. Their null hypothesis is H0: p=0.65, and based on a sample they find a sample proportion of 0.70 and a p-value of 0.053. Is there convincing statistical evidence at the 0.05 level of significance that the true proportion of individuals who use the competitor's card is actually greater than 0.65 ?

No, because the pp-value 0.053 is greater than the significance level 0.05.

A city planner wants to estimate the proportion of city residents who commute to work by subway each day. A random sample of 30 city residents was selected, and 28 of those selected indicated that they rode the subway to work. Is it appropriate to assume that the sampling distribution of the sample proportion is approximately normal?

No, because the sample is not large enough to satisfy the normality conditions.

The following list shows three factors that can either increase or decrease the probability of a Type II error. The sample size is increased. The significance level is increased. The standard error is increased. Which factors alone will cause the probability of a Type II error to increase?

III only An increase in standard error increases variability of the sampling distribution. Due to the increase in variability of the sampling distribution, it is more difficult to detect a true difference, and the probability of a Type IIII error is increased. An increase in sample size will increase power and decrease the probability of a Type IIII error. An increase in significance level will increase the probability of a Type II error and decrease the probability of a Type IIII error.

Market researchers selected a random sample of people from region A and a random sample of people from region B. The researchers asked the people in the samples whether they had tried a new product. The difference between the sample proportions (B minus A) of people in the regions who indicated they had tried the new product was 0.15. Under the assumption that all conditions for inference were met, a hypothesis test was conducted with the alternative hypothesis being that the population proportion of B is greater than that of A. The p-value of the test was 0.34. Which of the following is the correct interpretation of the p-value?

If the proportions of all people who have tried the new product is the same for both regions, the probability of observing a difference of at least 0.15 is 0.34.

Lila and Robert attend different high schools. They will estimate the population percentage of students at their respective schools who have seen a certain movie. Lila and Robert each select a random sample of students from their respective schools and use the data to create a 95 percent confidence interval. Lila's interval is (0.30,0.35), and Robert's interval is (0.27,0.34). Which of the following statements can be concluded from the intervals?

Lila's sample size is most likely greater than Robert's sample size. Wider intervals are usually the result of smaller sample sizes, not larger sample sizes.

A service company recently revised its call-routing procedures in an attempt to increase efficiency in routing customer calls to the appropriate agents. A random sample of customer calls was taken before the revision, and another random sample of customer calls was taken after the revision. The selected customers were asked if they were satisfied with the service call. The difference in the proportions of customers who indicated they were satisfied (pafter−pbefore) was calculated. A 90 percent confidence interval for the difference is given as (−0.02,0.11). The manager of the company claims that the revision in procedure will change the proportion of customers who will indicate satisfaction with their calls. Does the confidence interval support the manager's claim?

No. The value of 0 is contained in the interval, which indicates that it is plausible that there is no difference in the proportion of customers who will indicate satisfaction with their calls. A value of 0 contained in a confidence interval for a difference indicates that 0 is a plausible value for that difference. It is plausible that the proportion of satisfied customers after the revision is the same as that before the revision. Knowing that there are more positive values in the interval than negative values does not provide information to assess the manager's claim.