Exam 7: Estimates and Sample Sizes

Determine whether the given conditions justify using the margin of error $\mathrm { E } = \mathrm { z } _ { \alpha / 2 } \sigma / \sqrt { \mathrm { n } }$ when finding a confidence interval estimate of the population mean $\mu$ . -The sample size is $\mathrm { n } = 10$ and $\sigma$ is not known.

Use the confidence level and sample data to find a confidence interval for estimating the population µ. -A laboratory tested 90 chicken eggs and found that the mean amount of cholesterol was 230 milligrams with $\sigma = 16.0$ milligrams. Construct a 95 percent confidence interval for the true mean cholesterol content, $\mu$ , of all such eggs.

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 -A $99 \%$ confidence interval (in inches) for the mean height of a population is $65.64 < \mu < 67.16$ . This result is pased on a sample of size 144 . If the confidence interval $65.85 < \mu < 66.95$ 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).

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

Find the margin of error. -The football coach randomly selected ten players and timed how long each player took to perform a certain drill. The times (in minutes)were: 5.5 5.7 5.0 5.9 5.3 5.9 5.5 5.9 5.0 5.5 Determine a 95 percent confidence interval for the mean time for all players.

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. -The mean replacement time for a random sample of 20 washing machines is 10.2 years and the standard deviation is 2.6 years. Construct a 99% confidence interval for the standard deviation, ?, of the replacement times of all washing machines of this type.

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p -Margin of error: $0.01$ ; confidence level: $95 \%$ ; from a prior study, $\hat { \mathrm { p } }$ is estimated by the decimal equivalent of $69 \%$

Find the margin of error for the 95% confidence interval used to estimate the population proportion -In a survey of 4100 T.V. viewers, 20% said they watch network news programs.

Provide an appropriate response. -Identify the correct distribution (z, t, or neither)for each of the following. Sample Sire Standard deviation Shape of the distribution z or or neither n=35 s=4.5 Somewhat skewed n=20 s=4.5 Bell shaped n=25 \sigma=4.5 Bell shaped n=20 \sigma=4.5 Extremely skewed

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p -Margin of error: $0.04$ ; confidence level: $95 \%$ ; from a prior study, $\hat { \mathrm { p } }$ is estimated by the decimal equivalent of $92 \%$ ,

Provide an appropriate response. - $E = z _ { \alpha / 2 } \cdot \frac { \sigma } { \sqrt { n } }$ $\text { or } \mathrm { E } = \mathrm { t } _ { \alpha / 2 } \cdot \frac { \mathrm { s } } { \sqrt { \mathrm { n } } } ?$

Find the margin of error for the 95% confidence interval used to estimate the population proportion -n = 102, x = 52; 88 percent

Find the margin of error. -A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 193 milligrams with s = 15.4 milligrams. Construct a 95 percent confidence interval for the true mean cholesterol content of all such eggs.

Find the margin of error for the 95% confidence interval used to estimate the population proportion -n = 84, x = 37; 98 percent

Provide an appropriate response. -Define margin of error. Explain the relation between the confidence interval and the error estimate. Suppose a confidence interval is $9.65 < \mu < 11.35$ . Find the sample mean $\bar { x }$ and the error estimate E.

Provide an appropriate response. -Define a point estimate. What is the best point estimate for $\mu ?$

Provide an appropriate response. -When determining sample size we need to know $\hat { p }$ If we have no prior information, what are two methods that can be used?

Use the confidence level and sample data to find the margin of error E -Systolic blood pressures for women aged $18 - 24 : 94 \%$ confidence; $n = 97 , \bar { x } = 114.8 \mathrm {~mm} \mathrm { Hg } , \sigma = 12.7 \mathrm {~mm} \mathrm { Hg }$

Use the margin of error, confidence level, and standard deviation ? to find the minimum sample size required to estimate an unknown population mean µ -Margin of error: $121, confidence level: 95%, $95 \% , \sigma = \$ 528$

Solve the problem. -30 randomly picked people were asked if they rented or owned their own home, 30 said they rented. Obtain a point estimate of the true proportion of home owners.