It is desired to test $H_0: \mu=75$ against $H_{\mathrm{a}}: \mu<75$ using $\alpha=.10$. The population in question is uniformly distributed with standard deviation 15. A random sample of size 49 will be drawn from the population. Describe the (approximate) sampling distribution of $\bar{x}$ under the assumption that $H_0$ is true.

David's phone has about 10,000 songs. The distribution of play time for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds.

Describe the shape of the sampling distribution of $\bar{x}$ for SRSs of size n=100 from the population of songs on David's phone. Justify your answer.

A random sample of 40 observations is to be drawn from a large population of measurements. It is known that 30% of the measurements in the population are 1’s, 20% are 2’s, 20% are 3’s, and 30% are 4’s. Give the mean and standard deviation of the (repeated) sampling distribution of

$$\overline { x }$$

the sample mean of the 40 observations.

David's iPod has about 10,000 songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Describe the shape of the sampling distribution of $\bar{x}$ for SRSs of size n=100 from the population of songs on David's iPod. Justify your answer.

Decide whether you should use a normal sampling distribution or a t-sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim.

A car company says that the mean gas mileage for its luxury sedan is at least 21 miles per gallon (mpg). You believe the claim is incorrect and find that a random sample of five cars has a mean gas mileage of 19 mpg and a standard deviation of 4 mpg. Assume the gas mileage of all of the company 's luxury sedans is normally distributed. At $\alpha$ = 0.05, test the company's claim.

David's iPod has about 10,000 songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Describe the shape of the sampling distribution of $\bar{x}$ for SRSs of size n=5 from the population of songs on David's iPod. Justify your answer.

Suppose the Spontanversation sampling problem for which individuals who have downloaded Spontanversation use the app an average of 30 times per day with a standard deviation of 6.

a. What is the possibility that $\bar{x}$ is within $.5$ of the population mean if a random sample of 50 individuals who have downloaded Spontanversation is used?

Decide whether the sampling method could result in a biased sample. Explain your reasoning.

On the first day of school, all of the incoming freshmen attend an orientation program. The principal wants to learn the opinions of the freshmen regarding the orientation program. He decides to ask the first 25 freshmen that he sees.

After finishing their meal, every fifth person that left the restaurant was surveyed about whether they ordered dessert after their meals. Out of 20 people, 12 said yes. Is this sampling method valid? If so, about how many of the 380 people who had dinner at the restaurant ate dessert?

A simple random sample of size n=1460 is obtained from a population whose size is N=1.500.000 and whose population proportion with a specified characteristic is p=0.42. (a) Describe the sampling distribution of $\hat{p}$. (b) What is the probability of obtaining x=657 or more individuals with the characteristic? (c) What is the probability of obtaining x=584 or fewer individuals with the characteristic?

In this lesson, you learned that the formula for the standard deviation of the sampling distribution of $\bar{x}$ is valid when the sample size is less than 10% of the population size. What if the sample size is larger than 10% of the population size? In this case, we use the finite population correction factor in the formula for the standard deviation, where N is the population size:

$$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}$$

Imagine that you are choosing an SRS from a population of size N=1000 where $\mu$=60 and $\sigma$=5.

Calculate the standard deviation of the sampling distribution of $\bar{x}$ for samples of size n=1000 using the finite population correction factor. Why does the value of the standard deviation make sense in this situation?

Administrators at your school want to know if more vegetarian items should be added to the lunch menu. They want to find out how many students are vegetarians. Give an example of one sampling method that could result in a biased sample, and one that is not likely to result in a biased sample. Explain your reasoning.

Suppose that cars arrive at Burger King's drive-through at the rate of 20 cars every hour between 12:00 noon and 1:00 P.M. A random sample of 40 one-hour time periods between 12:00 noon and 1:00 P.M. is selected and has 22.1 as the mean number of cars arriving. (a) Why is the sampling distribution of $\bar{x}$ approximately normal? (b) What is the mean and standard deviation of the sampling distribution of $\overline{\mathbf{x}}$ assuming that $\mu$=20 and $\sigma=\sqrt{20} ?$ (c) What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at least 22.1 cars? Is this result unusual? What might we conclude?

Finding Probabilities for Electronic Associates, Inc., Managers. In the EAI sampling problem (see Figure 7.8), we showed that for $n=30$, there was $.5064$ probability of obtaining a sample mean within $\pm \$ 500$ of the population mean.

a. What is the probability that $\bar{x}$ is within $\$ 500$ of the population mean if a sample of size 60 is used?