Exam 12: Inferences for Population Proportions

According to a magazine article, 34% of married men in the U.S. are dissatisfied with their marriage. Identify the specified attribute.

A magazine poll of unemployed men in the U.S. stated "22% of those polled suffer from clinical depression; the margin of error for the poll is plus or minus 6 percentage points." How would you Interpret this statement? Assume that the margin of error is associated with a 95% confidence Interval.

A researcher is interested in estimating the proportion of adults in the U.S. who suffer from a rare form of cancer. In a random sample of 1000 adults in the U.S. she finds that 0.3% suffer from this form of cancer. She then obtains the following 90% confidence interval: $0.003 \pm 1.645 \sqrt { ( 0.003 ) ( 0.997 ) / 1000 }$ or 0.000155 to 0.0058 She concludes that she can be 90% confident that the true proportion of adults in the U.S. suffering from this form of cancer is somewhere between 0.0155% and 0.58%. Is anything wrong with this reasoning? Explain your answer.

A hypothesis test is to be performed for a population proportion. For the given sample data and null hypothesis, compute the value of the test statistic, $z = \frac { \hat { p } - \mathrm { p } _ { 0 } } { \sqrt { { p } _ { 0 } \left( 1 - \mathrm { p } _ { 0 } \right) / \mathrm { n } } }$ -A research group wants to determine whether the proportion of car accidents caused by drivers using cell phones has changed from the previous value of 13%. They obtained 10,000 auto accident Reports and found that 14% were caused by drivers using cell phones. The hypotheses are $\mathrm { H } _ { 0 } : \mathrm { p } = 0.13 , \mathrm { H } _ { \mathrm { a } } : \mathrm { p } \neq 0.13$ where p is the proportion of car accidents caused by drivers using cell Phones.

Find the indicated margin of error. -In a survey of 9500 T.V. viewers, 2850 said they watch network news programs. Find the margin of error for the 99% confidence interval used to estimate the population proportion.

A two-proportions z-test is to be performed using the P-value approach. Use the given sample data to find the P-valuefor the hypothesis test. - $\mathrm { x } _ { 1 } = 41 , \mathrm { n } _ { 1 } = 100 , \mathrm { x } _ { 2 } = 35 , \mathrm { n } _ { 2 } = 140 ; \mathrm { H } _ { 0 } : \mathrm { p } _ { 1 } = \mathrm { p } _ { 2 } , \mathrm { H } _ { \mathrm { a } } : \mathrm { p } _ { 1 } > \mathrm { p } _ { 2 } , \alpha = 0.10$

A mayoral election race is tightly contested. In a random sample of 1500 likely voters, 780 said that they were planning to vote for the current mayor. Based on this sample, would you claim that the mayor will win a majority of the votes? Explain.

Use the two-proportions z-test to perform the required hypothesis test. Assume that independent simple randomsamples have been selected from the two populations. Use the critical-value approach. -7 of 8500 people vaccinated against a certain disease later developed the disease. 18 of 10,000 people vaccinated with a placebo later developed the disease. At the 2% significance level, do the data provide sufficient evidence to conclude that those who are vaccinated against the disease are at lower risk of developing the disease than those vaccinated with a placebo?

Assume that you wish to estimate a population proportion, p. For the given margin of error and confidence level,determine the sample size required. -A political action committee is interested in finding out what proportion of voters will support an environmental initiative. Obtain a sample size that will ensure a margin of error of at most 0.07 for A 95% confidence interval. Similar initiatives in the past have gotten 94% support.

Suppose that a population consists of 30 men and 50 women. Let $\hat { p }$ represent the proportion of women in a random sample of size 10 from this population. What is the mean of $\hat { p }$ for all possible samples of size 10 from this population?

Use the two-proportions z-test to perform the required hypothesis test. Assume that independent simple randomsamples have been selected from the two populations. Use the critical-value approach. -A marketing survey involves product recognition in New York and California. Of 558 New Yorkers surveyed, 193 recognized the product while 196 out of 614 Californians recognized the product. At the 5% significance level, do the data provide sufficient evidence to conclude that the recognition rate in New York differs from the recognition rate in California?

Use the one-proportion z-test to perform the required hypothesis test. Use the critical-value approach. -A poll of 1,068 adult Americans reveals that 48% of the voters surveyed prefer the Democratic candidate for the presidency. At the 0.05 level of significance, do the data provide sufficient evidence that the percentage of all voters who prefer the Democrat is less than 50%?

A researcher wishes to test whether there is a difference between the proportions of women and men who favor stricter gun control legislation. In a random sample of 300 women, 69% favored stricter gun control legislation. In a random sample of 200 men, 58% favored stricter gun control legislation. Identify the specified attribute. Determine, if possible, the two sample proportions. If it is not possible to determine the sample proportions, explain why not. Determine, if possible, the two population proportions. If it is not possible to determine the population proportions, explain why not.

Use the two-proportions z-test to perform the required hypothesis test. Assume that independent simple randomsamples have been selected from the two populations. Use the P-value approach. -Of 85 randomly selected women, 38 have tried some form of alternative medicine. Of 90 randomly selected men, 23 have tried some form of alternative medicine. At the 1% significance level, do the data provide sufficient evidence to conclude that women are more likely than men to try alternative forms of medicine?

The numbers of successes and the sample sizes are given for independent simple random samples from two populations.Decide whether using the two-proportions z-procedures is appropriate. - $\mathrm { x } _ { 1 } = 18 , \mathrm { n } _ { 1 } = 30 , \mathrm { x } _ { 2 } = 12 , \mathrm { n } _ { 2 } = 30$

Use the two-proportions z-test to perform the required hypothesis test. Assume that independent simple randomsamples have been selected from the two populations. Use the critical-value approach. - Use the given sample data to test the claim that < . Use a significance level of 0.10 . =84=95

The number of successes and the sample size are given for a simple random sample from a population. Decide whetherusing the one-proportion z-interval procedure is appropriate. -x = 86, n = 90

The number of successes and the sample size are given for a simple random sample from a population. Decide whetherusing the one-proportion z-test is appropriate. - $x = 60 , n = 120 , H _ { 0 } : p = 0.6 , H _ { a } : p < 0.6$

According to a magazine article, 34% of married men in the U.S. are dissatisfied with their marriage. Under what circumstances is the proportion 0.34 a population proportion? a sample proportion?

Provide an appropriate response. -For what kinds of data might you be interested in conducting inferences for a population proportion and for what kinds of data might you be interested in conducting inferences for a population mean? Explain how you distinguish the two types of data and give an example of each type of data.