7. Sampling Distributions & Confidence Intervals: Mean

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?

c=0.95, x̅=3.46, s=1.63, n=16

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?

c=0.90, x̅=8.21, σ=0.62, n=8

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)

48,69446,85642,91261,67271,11254,861

69,45471,84159,75169,61254,28452,166

66,36048,16465,27235,25061,12765,397

58,92558,91659,01753,07045,19969,941

69,49257,08553,82952,69268,29853,792

Construct a 95% confidence interval for the population mean annual earnings for locksmiths.

Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

Renewable Energy During a recent period of two years, the day-ahead prices for renewable energy in Germany (in euros per mega-watt hour) have a mean of 31.58 and a standard deviation of 12.293. Random samples of size 75 are drawn from this population, and the mean of each sample is determined.

Repeat Exercise 26 for samples of size 72 and 108. What happens to the mean and the standard deviation of the distribution of sample means as the sample size increases?

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 $\mu$.