Exam 8: Confidence Intervals for One Population Mean

Write the word or phrase that best completes each statement or answers the question. -Rank the following t- and z-values in order from smallest to largest. (a) $\mathrm { Z } _ { 0.05 }$ (b) $\mathrm { t } _ { 0.05 }$ (c) $t _ { 0.05 }$ (d) $\mathrm { t } _ { 0.95 }$ (e) $\mathrm { t } _ { 0.01 }$ The degrees of freedom in (b), (d), (e)are identical. The degrees of freedom in (c)are greater than those in (b), (d), (e).

Find the confidence interval specified. -A sample of 38 people were randomly selected from among the workers in a shoe factory. The time taken for each person to polish a finished shoe was measured. The sample mean was 4.0 minutes. Assume that $\sigma =$ 0.71 minutes. Construct a 90% confidence interval for the true mean time, $\mu$ to Polish a shoe.

Find the requested value. -A researcher for a car insurance company wishes to estimate the mean annual premium that men aged 20-24 pay for their car insurance. A random sample of 16 men aged between 20 and 24 yields The following annual premiums, in dollars. 616 421 736 948 510 966 948 439 585 737 580 725 856 610 720 985 Use the data to obtain a point estimate of the mean annual premium for all men aged between 20 And 24. Round your answer to the nearest dollar.

Find the value of α that corresponds to a confidence level of 86.5%.

Solve the problem. -Based on a sample of 36 randomly selected years, a 90% confidence interval for the mean annual precipitation in one city is from 46.8 inches to 49.2 inches. Find the margin of error.

Find the confidence interval specified. -30 students were randomly selected from a large group of students taking a certain calculus test. The mean score for the students in the sample was 78. Assume that $\sigma = 14.5$ . Construct a $99 \%$ confidence interval for the mean score, $\mu$ , of all students taking the test.

Find the specified t-value. -For a t-curve with $\mathrm { df } = 9$ , find the t-value having area $0.005$ to its left.

Find the confidence interval specified. Assume that the population is normally distributed. -The principal randomly selected six students to take an aptitude test. Their scores were: 87.4 86.9 89.9 78.3 75.1 70.6 Determine a 90% confidence interval for the mean score for all students.

Solve the problem. -A sample of 55 eggs yields a mean weight of 1.76 ounces. Assuming that $\sigma$ = 0.28 ounces, find the margin of error in estimating µ at the 95% level of confidence.

Solve the problem. -A confidence interval for a population mean has a margin of error of 6.4. Determine the length of the confidence interval.

Find the necessary sample size. -The weekly earnings of students in one age group are normally distributed with a standard deviation of 10 dollars. A researcher wishes to estimate the mean weekly earnings of students in This age group. Find the sample size needed to assure with 95 percent confidence that the sample Mean will not differ from the population mean by more than 2 dollars.

Write the word or phrase that best completes each statement or answers the question. -Based on a sample of size 25, a researcher obtains an estimate of 64.4 inches for the mean height of all women aged 30-40. At the 95% confidence level, the margin of error is 1.2 inches. Do you agree with the interpretation below? If not, explain why not and give a correct interpretation. Researcher's interpretation: "We can be 95% confident that the height of a randomly selected woman will differ from 64.4 inches by at most 1.2 inches."

Write the word or phrase that best completes each statement or answers the question. -A pizza house is considering offering the following special: "Pizza delivered in less than 30 minutes or the pizza is free." Based on a random sample of 40 delivery times, the manager obtains the following 99% confidence interval for the true mean delivery time, $\mu :$ : 25 minutes to 30 minutes. The manufacturer concludes that at least 99% of the pizzas are delivered in under 30 minutes and that it will be safe to go ahead with the special offer. Is the manager's interpretation correct? Why or why not? Explain your thinking.

Suppose that you wish to obtain a confidence interval for a population mean. Under the conditions described below, should you use the z-interval procedure, the t-interval procedure, or neither? - The population standard deviation is known. - The population is far from being normally distributed. - the sample size is large.

Write the word or phrase that best completes each statement or answers the question. -A coffee manufacturer claims that the mean amount of coffee in its 4-ounce jars is 4.1 ounces. Based on a sample of 50 jars, a consumer advocate group obtains the following 99% confidence interval for the population mean weight, $\mu _ { i }$ 3.95 ounces to 4.05 ounces. Based on this interval, do you think that the manufacturer's claim is plausible? Possible? Explain your thinking.

Find the requested confidence interval. -A college statistics professor has office hours from 9:00 A.M. to $10 : 30$ A.M. daily. A sample of waiting times to see the professor (in minutes) is $10,12,20,15,17,10,30,28,35,28,19,27,25,22,33$ , $37,14,21,20,23$ . Assuming $\sigma = 7.84$ , find the $99.74 \%$ confidence interval for the population mean.

Provide an appropriate response. -Suppose you have obtained a 95% confidence interval for $\mu .$ Which of the following statements is/are true regarding the relationship between precision and confidence level? Assume that the Sample size is fixed. A. Increasing the confidence level to 99% will result in a narrower interval. B. Decreasing the confidence level to 90% will result in greater precision. C. Decreasing the precision will result in a higher confidence level. D. Increasing the precision will result in a higher confidence level.

Write the word or phrase that best completes each statement or answers the question. -A researcher estimates that the mean systolic blood pressure for women aged between 18 and 24 is 118 mmHg. At the 95% confidence level, the margin of error is 2.4 mmHg. How would you interpret this margin of error?

Find the necessary sample size. -Scores on a certain test are normally distributed with a variance of 68. A researcher wishes to estimate the mean score achieved by all adults on the test. Find the sample size needed to assure With 95 percent confidence that the sample mean will not differ from the population mean by more Than 4 units.

A confidence interval for a population mean, µ, has length $\mu ,$ 86. Find the margin of error.