Exam 7: Estimates and Sample Sizes

Solve the problem. -A one-sided confidence interval for $\mathrm { p }$ can be written as $\mathrm { p } < \hat { \mathrm { p } } + \mathrm { E }$ or $\mathrm { p } > \hat { \mathrm { p } } - \mathrm { E }$ where the margin of error $\mathrm { E }$ is modified by replacing $\mathrm { z } _ { \alpha / 2 }$ with $\mathrm { z } _ { \alpha }$ . If a teacher wants to report that the fail rate on a test is at most $x$ with $90 \%$ confidence, construct the appropriate one-sided confidence interval. Assume that a simple random sample of 58 students results in 5 who fail the test.

Express the confidence interval in the form of $\hat{p}$ ± E. - $- 0.128 < p < 1.212$

Provide an appropriate response. -Describe the steps for finding a confidence interval.

Find the margin of error for the 95% confidence interval used to estimate the population proportion -n = 230, x = 90

Express the confidence interval in the form of $\hat{p}$ ± E. -0.02 < p < 0.48

Solve the problem. -Find the point estimate of the true proportion of people who wear hearing aids if, in a random sample of 875 people, 56 people had hearing aids.

Provide an appropriate response. -When determining the sample size for a desired margin of error, the formula is $\mathrm{n}=\frac{\left[{ }^{\mathrm{z}} \alpha / 2\right]^{2} \cdot \mathrm{\hat{p}\hat{q}}}{\mathrm{E}^{2}}$ . Based on this formula, discuss the fact that sample size is not dependent on the population size; that is, it is not necessary to sample a particular percent of the population.

Why would manufacturers and businesses be interested in constructing a confidence interval for the population variance? Would manufacturers and businesses want large or small variances?

A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct the 95% confidence interval for the standard deviation, ?, of the scores of all subjects.

Solve the problem. -A researcher wishes to construct a 95% confidence interval for a population mean. She selects a simple random sample of size n = 20 from the population. The population is normally distributed and is known. When constructing the confidence interval, the researcher should use the normal distribution; however, she incorrectly uses the t distribution. How does this incorrectly calculated confidence relate to the correct confidence interval?

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p -Margin of error: $0.007$ ; confidence level: $99 \%$ ; from a prior study, $\hat { \mathrm { p } }$ is estimated by $0.238$

Provide an appropriate response. -What assumption about the parent population is needed to use the t distribution to compute the margin of error?

Solve the problem. -A researcher wishes to construct a 95% confidence interval for a population mean. She selects a simple random sample of size n = 20 from the population. The population is normally distributed and is unknown. When constructing the confidence interval, the researcher should use the t distribution; however, she incorrectly uses the normal distribution. Will the true confidence level of the resulting confidence interval be greater than 95%, smaller than 95%, or exactly 95%?

Find the appropriate minimum sample size -You want to be 95% confident that the sample variance is within 20% of the true population variance.

Find the margin of error. -A savings and loan association needs information concerning the checking account balances of its local customers. A random sample of 14 accounts was checked and yielded a mean balance of $664.14 and a standard deviation of $297.29. Find a 98% confidence interval for the true mean checking account balance for local customers.

Solve the problem. -A simple random sample of students is selected, and the students are asked how much time they spent preparing for a test. The times (in hours)are as follows: 1.3 7.2 4.2 12.5 6.6 2.5 5.5 Based on these results, a confidence interval for the population mean is found to be $\mu = 5.7 \pm 4.4$ . Find the degree confidence.

Do one of the following, as appropriate: (a) Find the critical value $z_{?/2}$ , (b) find the critical value $t_{?/2}$ , (c) state that neither the normal nor the t distribution applies - $90 \% ; \mathrm { n } = 10 ; \sigma$ is unknown; population appears to be normally distributed.

Provide an appropriate response. -Describe the process for finding the confidence interval for a population proportion.

Solve the problem. -A researcher is interested in estimating the proportion of voters who favor a tax on e-commerce. Based on a sample of 250 people, she obtains the following 99% confidence interval for the population proportion p: 0.113 < p < 0.171 Which of the statements below is a valid interpretation of this confidence interval? A: There is a 99% chance that the true value of p lies between 0.113 and 0.171. B: If many different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, 99% of the time the true value of p would lie between 0.113 and 0.171. C: If many different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, in the long run 99% of the confidence intervals would contain the true value of p. D: If 100 different samples of size 250 were selected and, based on each sample, a confidence interval were constructed, exactly 99 of these confidence intervals would contain the true value of p.

Find the margin of error. -The daily intakes of milk (in ounces) for ten randomly selected people were: 19.8 15.4 25.5 15.5 20.5 24.6 15.6 31.0 28.0 21.4 Find a 99 percent confidence interval for the population standard deviation $\sigma$ .