Exam 7: Estimates and Sample Size

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. - $95 \% \text { confidence; } n = 2015 , x = 1631$

Find the critical value $z _ { \alpha / 2 }$ that corresponds to a 91% confidence level.

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. - $\mathrm { n } = 171 , \mathrm { x } = 124 ; 95 \%$ confidence

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. - $90 \% \text { confidence; } n = 430 , x = 80$

Use the confidence level and sample data to find a confidence interval for estimating the population µ. Round your answer to the same number of decimal places as the sample mean. -39 packages are randomly selected from packages received by a parcel service. The sample has a mean weight of 29.9 pounds and a standard deviation of 3.0 pounds. What is the 95% confidence interval for the true mean weight, µ, of all packages received by the parcel service?

Use the given degree of confidence and sample data to construct a confidence interval for the population mean µ. Assume that the population has a normal distribution. -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.

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. -When 334 college students are randomly selected and surveyed, it is found that 103 own a car. Find a 99% confidence interval for the true proportion of all college students who own a car.

A $99 \%$ confidence interval (in inches) for the mean height of a population is $65.33 < \mu < 66.87$ . This result is based on a sample of size 144 . If the confidence interval $65.54 < \mu < 66.66$ is obtained from the same sample data, what is the degree of confidence? (Hint: you will first need to find the sample mean and sample standard deviation).

In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$ , a table is used to find the critical values $\chi { } _ { L } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ for values of $\mathrm { n } \leq 101$ . For larger values of $n , \chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ can be approximated by using the following formula: $\chi ^ { 2 } = \frac { 1 } { 2 } \left[ \pm \mathrm { z } _ { \alpha / 2 } + \sqrt { 2 \mathrm { k } - 1 } \right] ^ { 2 }$ where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z } _ { \alpha / 2 }$ is the critical z score. Estimate the critical values $\chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ for a situation in which you wish to construct a $95 \%$ confidence interval for $\sigma$ and in which the sample size is $n = 295$ .

The confidence interval, $4.40 < \sigma < 6.39$ , for the population standard deviation is based on the following sample statistics: $\mathrm { n } = 41 , \overline { \mathrm { x } } = 42.1$ , and $\mathrm { s } = 5.2$ . What is the degree of confidence?

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. -99% confidence; n $\mathrm { n } = 6400 , \mathrm { x } = 1920$

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. -In a random sample of 187 college students, 106 had part-time jobs. Find the margin of error for the 95% confidence interval used to estimate the population proportion.

Explain how confidence intervals might be used to make decisions. Give an example to clarify your explanation.

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. -Of 125 adults selected randomly from one town, 31 of them smoke. Construct a 99% confidence interval for the true percentage of all adults in the town that smoke.

Suppose we wish to construct a confidence interval for a population proportion p. If we sample without replacement from a relatively small population of size N, the margin of error E is modified to include the finite population correction factor as follows: $E = z _ { \alpha } / 2 \sqrt { \frac { \hat { p q } } { n } } \sqrt { \frac { N - n } { N - 1 } }$ Construct a $90 \%$ confidence interval for the proportion of students at a school who are left handed. The number of students at the school is $N = 410$ . In a random sample of 89 students, selected without replacement, there are 10 left handers.

Assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. Round the margin of error to four decimal places. -99% confidence; the sample size is 1180, of which 45% are successes

Use the given degree of confidence and sample data to find a confidence interval for the population standard deviation ?. Assume that the population has a normal distribution. Round the confidence interval limits to the same number of decimal places as the sample standard deviation. -The mean replacement time for a random sample of 20 washing machines is 9.4 years and the standard deviation is 2.5 years. Construct a 99% confidence interval for the standard deviation, ?, of the replacement times of all washing machines of this type.

Find the value of $- \mathrm { z } _ { \alpha / 2 }$ that corresponds to a confidence level of 97.80%.

Use the given information to find the minimum sample size required to estimate an unknown population mean µ. -How many weeks of data must be randomly sampled to estimate the mean weekly sales of a new line of athletic footwear? We want 99% confidence that the sample mean is within $200 of the population mean, and the population standard deviation is known to be $1400.

Use the given data to find the minimum sample size required to estimate the population proportion. -Margin of error: 0.04; confidence level: 99%; from a prior study, $\hat { p }$ is estimated by 0.14.