Confidence Intervals for One Population Mean

Suppose that you take $500$ simple random samples from a population and that, for each sample, you obtain a $90\%$ confidence interval for an unknown parameter. Approximately how many of those confidence intervals will not contain the value of the unknown parameter?

Suppose that you take $1000$ simple random samples from a population and that, for each sample, you obtain a 95% confidence interval for an unknown parameter. Approximately how many of those confidence intervals will contain the value of the unknown parameter?

Express the form of most of the confidence intervals that you will encounter in your study of statistics in terms of "point estimate" and "margin of error." 

When estimating an unknown parameter, what does the margin of error indicate?

What is a confidence interval estimate of a parameter? Why is such an estimate superior to a point estimate?

The value of a statistic used to estimate a parameter is called a ______ of the parameter.

A simple randoes sample is taken from a population and yields the following data for a variable of the population:

Find a point estimatate for the population mean.(i.e, the mean of the variable)

A simple random sample is taken from a population and yields the following data for a variable of the population:

Find a point estimate for the population mean (i.e., the mean of the variable).

A simple randoes sample is taken from a population and yields the following data for a variable of the population: 

 find a point estimate for the population standard deviation (i.e., the standard deviation of the variable).

A simple random sample is taken from a population and yields the following data for a variable of the population: 

 find a point estimate for the population standard deviation (i.e., the standard deviation of the variable). 

We provide a sample mean, sample size, and population standard deviation. In each case, perform the following tasks.

a. Find a 95% confidence interval for the popularion mean. (Note: You may want to review Example 8.2, which begins on page 316.)

b. Identify and interpret the margin of error:

c. Express the endpoints of the confidence interval in terms of the point estimate and the margin of error.

where, $\hat{x}=20,n=36,\sigma =3$

In each of the Exercises $8.11-8.16$, we provide a sample mean, sample size, and population standard deviation. In each case, perform the following tasks.

a. Find a $95\%$ confidence interval for the population means. (Note: You may want to review Example 8.2 , which begins on page $316$

b. Identify and interpret the margin of error.

c. Express the endpoints of the confidence interval in terms of the point estimate and the margin of error.

$\hat{x}=25,n=36,\sigma =3$

In each Exercises $8.11-8.16$, we provide a sample mean, sample size, and population standard deviation. In each case, perform the following tasks.

a. Find a $95\%$ confidence interval for the population mean. (Note: You may want to review Example $8.2$, which begins on the page $316$)

b. Identify and interpret the margin of error.

c. Express the endpoints of the confidence interval in terms of the point estimate and the margin of error.

$\overline{x}=30,n=25,\sigma =4$

In each Exercises $8.11-8.16$, we provide a sample mean, sample size, and population standard deviation. In each case, perform the following Lacks.

a. Find a $95\%$ confidence interval for the population mean. (Note: You may want to review Example $8.2$, which begins on the page $316$)

b. Identify and interpret the margin of error.

c. Express the endpoints of the confidence interval in terms of the point estimate and the margin of error.

$\hat{x}=35, n=25, \sigma =4$

In each of Exercises $8.11-8.16$, we provide a sample mean, sample size, and population standard deviation. In each case, perform the following Lacks.

a. Find a $95\%$ confidence interval for the population means. (Note: You may want to review Example $8.2$, which begins on page $3.16$)

b. Identify and interpret the margin of error.

c. Express the endpoints of the confidence interval in terms of the point estimate and the margin of error.

$\overline{x}=50, n=16, \sigma =5$

In each Exercises $8.11-8.16$, we provide a sample mean, sample size, and population standard deviation. In each case, perform the following tasks.

a. Find a $95\%$ confidence interval for the population mean. (Note: You may want to review Example $8.2$, which begins on the page $3l6$)

b. Identify and interpret the margin of error.

c. Express the endpoints of the confidence interval in terms of the point estimate and the margin of error:

$\hat{x}=55,n=16,\sigma =5$

Wedding Costs. According to Bride's Magazine, getting married these days can be expensive when the costs of the reception, engagement ring, bridal gown, and pictures-just to name a few-are included. A simple random sample of $20$ recent U.S. weddings yielded the following data on wedding costs, in dollars. 

a. Use the data to obtain a point estimate for the population mean wedding cost. $\mu$, of all recent U.S. weddings. (Note: The sum of the data is $\$526,538$.)

b. Is your point estimate in part (a) likely to equal $\mu$exactly? Explain your answer.

Cottonmouth Litter Size. In the article "The Eastern Cottonmouth (Agkistrodon piscivorus) at the Northern Edge of Its Range" (Journal of Herpetology, Vol. $29$ No. $3$ pp. $391-398$), C. Blem and L. Blem examined the reproductive characteristics of the eastern cottonmouth, a once widely distributed snake whose numbers have decreased recently due to encroachment by humans. A simple random sample of $44$ female cottonmouths yielded the following data on the number of young per litter. 

a. Use the data to obtain a point estimate for the mean number of young per litter, $\mu$ of all female eastern cottonmouths. (Note: $\left.\Sigma x_i=334.\right)$

b. Is your point estimate in part (a) likely to equal $\mu$ exactly? Explain your answer.

Wedding Costs. Refer to Exercise $8.17.$ Assume that recent wedding costs in the United States are normally distributed with a standard deviation of $\$8100$

a. Determine a $95\%$ confidence interval for the mean cost. $\mu$, of all recent U.S. weddings.

b. Interpret your result in part (a).

c. Does the mean cost of all recent U.S. weddings lie in the confidence interval you obtained in part (a)? Explain your answer.

Cottonmouth Litter Size. Refer to Exercise 8.18. Assume that $\sigma =2.4$

a. Obtain a $95\%$ confidence interval for the mean number of young per litter of all-female eastern cottonmouths.

b. Interpret your result in part (a).

c. Does the mean number of a young per litter of all-female eastern cottonmouths lie in the confidence interval you obtained in part (a)? Explain your answer.

Fuel Tank Capacity. Consumer Reports provides information on new automobiles models-including price, mileage ratings, engine size, body size, and indicators of features. A simple random sample of $35$ new models yielded the following data on fuel tank capacity, in gallons. 

a. Determine a point estimate for the mean fuel tank capacity of all new automobile models. Interpret your answer in words. (Note: $\Sigma x_i=664.9$ gallons.)

b. Determine a $95\%$ confidence interval for the mean fuel tank capacity of all new automobile models. Assume $\sigma =3.50$ gallons.

c. How would you decide whether fuel tank capacities for new automobile models are approximately normally distributed?

d. Must fuel tank capacities for new automobile models be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? Explain your answer.

Home Improvements. The American Express Retail Index provides information on budget amounts for home improvements. The following table displays the budgets, in dollars, of 45 randomly sampled home improvement jobs in the United States.

a. Determine a point estimate for the population mean budget, $\mu$ for such home improvement jobs. Interpret your answer in words. (Note: The sum of the data is $\backslash (129,849$)

b. Obtain a $95\%$ confidence interval for the population mean budget $\mu$ for such home improvement jobs and interpret your result in words. Assume that the population standard deviation of budgets for home improvement jobs is $\backslash )1350$

c. How would you decide whether budgets for such home improvement jobs are approximately normally distributed?

d. Must the budgets for such home improvement jobs be exactly normally distributed for the confidence interval that you obtained in part (b) to be approximately correct? Explain your answer.

Giant Tarantulas. A tarantula has two body parts. The anterior part of the body is covered above by a shell. or carapace. In the paper "Reproductive Biology of Uruguayan Theraphosids" (The Journal of Arachnology, Vol. $30$, No. $3$ Pp. $571-587$), F. Costa and F. Perez-Miles discussed a large species of tarantula whose common name is the Brazilian giant tawny red. A simple random sample of $15$ of these adult male tarantulas provided the following data on carapace length, in millimeters $\left(\mathrm{mm}\right)$ 

a. Obtain a normal probability plot of the data.

b. Based on your result from part (a), is it reasonable to presume that the carapace length of adult male Brazilian giant tawny red tarantulas is normally distributed? Explain your answer.

c. Find and interpret a $95\%$ confidence interval for the mean carapace length of all adult male Brazilian giant tawny red tarantulas. The population standard deviation is $1.76 \mathrm{mm}$

d. In Exercise $6.97$ we noted that the mean carapace length of all adult male Brazilian giant tawny red tarantulas is \(18.14 \mathrm{~mm}\). Does your confidence interval in part (c) contain the population means? Would it necessarily have to? Explain your answers.

Serum Cholesterol Levels. Information on serum total cholesterol levels is published by the Centers for Disease Control and 

Prevention in National Health and Nutrition Examination Survey. A simple random sample of $12$ U.S. females $20$ years old or older provided the following data on serum total cholesterol level, in milligrams per deciliter (mg/dL).

a. Obtain a normal probability plot of the data.

b. Based on your result from part (a), is it reasonable to presume that serum total cholesterol level of U.S. females $20$ years old or older is normally distributed? Explain your answer.

c. Find and interpret a $95\%$ confidence interval for the mean serum total cholesterol level of U.S. females 20 years old or older. The population standard deviation is $44.7\mathrm{mg}/\mathrm{dL}$

d. In Exercise $6.98$, we noted that the mean serum total cholesterol level of U.S. females $20$ years old or older is $206\mathrm{mg}/\mathrm{dL}$ Does your confidence interval in part (c) contain the population means? Would it necessarily have to? Explain your answers.

New Mobile Homes. Refer to Examples 8.1 and 8.2. Use the data in Table \(8.1\) on page 315 to obtain a \(99.7 \%\) confidence interval for the mean price of all new mobile homes. (Hint: Proceed as in Example 8.2, but use Property 3 of the empirical rule on page 271 instead of Property 2.)

New Mobile Homes. Refer to Examples $8.1$ and $8.2$ Use the data in Table $8.1$ on page $315$ to obtain a $68\%$ confidence interval for the mean price of all new mobile homes. (Hint: Proceed as in Example $8.2$, but use Property $1$ of the empirical rule on page $271$ instead of Property $2$ .)

Explain why the margin of error determines the accuracy with which a sample mean estimates a population mean.

Suppose that you have obtained data by taking a random sample from a population. Before performing a statistical inference, what should you do?

Suppose that you have obtained data by taking a random sample from a population and that you intend to find a confidence interval for the population mean, $\mu$. Which confidence level, $95\%$ or $99\%$, will result in the confidence interval giving a more accurate estimate of $\mu$ ?

Suppose that you will be taking a random sample from a population and that you intend to find a $95\%$ confidence interval for the population mean $\mu$. Which sample size, $25 \mathrm{or} 30$ , will result in the confidence interval giving a more accurate estimate of $\mu$?

Discuss the relationship between the margin of error and the standard error of the mean.

Find the confidence level and $\propto$  for

a. $90\%$confidence interval.

b. $94\%$ confidence interval.

Find the confidence level and $\alpha$ for

a. $85\%$confidence interval.

b.  $95\%$confidence interval

What is meant by saying that a $1-\alpha$ confidence interval is

a. exact?

b. approximately correct?

For what is normal population an abbreviation?

What is meant by saying that a statistical procedure is robust?

Assume that the population standard deviation is known and decide weather use of the z-interval procedure to obtain a confidence interval for the population mean is reasonable. Explain your answers.

The variable under consideration is very close to being normally distributed, and the sample size is $10$ .

Assume that the population standard deviation is known and decide weather use of the $z$ interval procedure to obtain a confidence interval for the population mean is reasonable. Explain your answers.

The variable under consideration is very close to being normally distributed, and the sample size is $75$.

Assume that the population standard deviation is known and decide weather use of the $z$-interval procedure to obtain a confidence interval for the population mean is reasonable. Explain your answers.

The sample data contain outliers, and the sample size is $20$.

A confidence interval for a population mean has length $20$ .

a. Determine the margin of error.

b. If the sample mean is $60$ , obtain the confidence interval.

c. Construct a graph that illustrates your results.

In developing Procedure $8.1$ we assumed that the variable under consideration is normally distributed.

a. Explain why we needed that assumption.

b. Explain why the procedure yields an approximately correct confidence interval for large samples, regardless of the distribution of the variable under consideration.

Refer to Procedure $8.1.$

a. Explain in detail the assumptions required for using the $z-$interval procedure.

b. How important is the normality assumption? Explain your answer.

A confidence interval for a population mean has a margin of error of $0.047$

a. Determine the length of the confidence interval.

b. If the sample mean is $0.205$, obtain the confidence interval.

c. Construct a graph that illustrates your results.

A confidence interval for a population mean has a margin of error of $3.4$

a. Determine the length of the confidence interval.

b. If the sample mean is $52.8$, obtain the confidence interval.

c. Construct a graph that illustrates your results.

Explain the effect on the margin of error and hence the effect on the accuracy of estimating a population mean by a sample mean.

Decreasing the sample size while keeping the same confidence level.

Explain the effect on the margin of error and hence the effect on the accuracy of estimating a population mean by a sample mean. 

Increasing the confidence level while keeping same sample size.

Explain the effect on the margin of error and hence the effect on the accuracy of estimating a population mean by a sample mean. 

Decreasing the confidence level while keeping the same sample size.

Explain the effect on the margin of error and hence the effect on the accuracy of estimating a population mean by a sample mean. 

Increasing the sample size while keeping the same confidence level.

A confidence interval for a population mean has a length of $162.6$.

a. Determine the margin of error.

b. If the sample mean is $643.1$, determine the confidence interval.

c. Construct a graph that illustrates your results.

State True or False. Give Reasons for your answers.

The length of the confidence interval can be determined if you know only the margin of error.