7. Sampling Distributions & Confidence Intervals: Mean

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.95, s^2 = 11.56, n = 30

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

c. Construct a 99% confidence interval for the population standard deviation. Interpret the results.

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)

Construct a 99% confidence interval for the population variance.

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)

Construct a 99% confidence interval for the population standard deviation.

Fill out the table using a calculator and $n=30$.

Find the critical val. $\left(t_{\alpha /2}\right)$ for each confidence interval.

(A) Confidence Level = $95\%,n=35$

Find the critical val. $\left(t_{\alpha \text{⁄}2}\right)$ for each confidence interval.

(B) Confidence Level = $90\%,n=48$

Find each probability.

(A) $P(T>1.5),df=8$

Find each probability.

(B)

A nutritionist wants to estimate the average grams of protein in a brand of protein bars. She takes a random sample of 40 protein bars with $\bar{x}=210$ g & knows from prior data that $\sigma =12$. Make a 95% conf. int. for $\mu$.

To study the concentration of a particular pollutant (in parts per million) in a local river, an environmental scientist collects 32 water samples from random locations. They get $\bar{x}=3.87$ ppm & know from previous data that $\sigma =0.62$ ppm. Make a 99% conf. int. for the mean pollutant concentration.

A technician wants to estimate the average battery life of a new type of smart phone, so he tests 8 randomly selected phones & records the data below. Assuming battery life has a normal dist, make a 90% conf. int. for the mean battery life.

Age of Death-Row InmatesIn 2002, the mean age of an inmate on death row was 40.7 years, according to data from the U.S. Department of Justice. A sociologist wondered whether the mean age of a death-row inmate has changed since then. She randomly selects 32 death-row inmates and finds that their mean age is 38.9, with a standard deviation of 9.6. Construct a 95% confidence interval about the mean age. What does the interval imply?

What requirements must be satisfied in order to construct a confidence interval about a population mean?

[DATA] Family Size A random sample of 60 married couples who have been married 7 years was asked the number of children they have. The results of the survey are as follows:

Note: x̄ = 2.27, s = 1.22.

b. Compute a 95% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval.