91影视

Tests and confidence intervals The P-value for a two-sided test of the null hypothesis $H_0:渭=15is0.03$

a. Does the $99\%$ confidence interval for $渭$ include $15$? Why or why not?

b. Does the $95\%$ confidence interval for $渭$ include $15$? Why or why not?

Potato chips A company that makes potato chips requires each shipment of

potatoes to meet certain quality standards. If the company finds convincing evidence that more than $8\%$of the potatoes in the shipment have 鈥渂lemishes,鈥� the truck will be sent back to the supplier to get another load of potatoes. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of potatoes from the shipment. He will then perform a test of $H_0:p=0.08$versus $H_a:p>0.08$, where p is the true proportion of potatoes with blemishes in a given truckload. The power of the test to detect that $p=0.11$, based on a random sample of $500$ potatoes and significance level $伪=0.05$, is $0.764$Interpret this value.

Upscale restaurant You are thinking about opening a restaurant and are searching for a good location. From the research you have done, you know that the mean income of those living near the restaurant must be over $\backslash (85,000$to support the type of upscale restaurant you wish to open. You decide to take a simple random sample of $50$people living near one potential site. Based on the mean income of this sample, you will perform a test at the

$伪=0.05$ significance level of $H_0:渭=\backslash )85,000$versus $H_a:渭>\backslash (85,000$, where $渭$ is the true mean income in the population of people who live near the restaurant. The power of the test to detect that $渭=\backslash )86,000$is $0.64$ Interpret this value.

Powerful potatoes Refer to Exercise 85. Determine if each of the following

changes would increase or decrease the power of the test. Explain your answers.

a. Change the significance level to $伪=0.10$

b. Take a random sample of $250$ potatoes instead of $500$ potatoes.

c. The true proportion is $p=0.10$ instead of $p=0.11$

Restaurant power Refer to Exercise $86$ Determine if each of the following changes would increase or decrease the power of the test. Explain your answers.

a. Use a random sample of $30$ people instead of $50$ people.

b. Try to detect that $渭=\backslash (85,500$ instead of $渭=\backslash )86,000$

c. Change the significance level to $伪=0.10$

Potato power problems Refer to Exercises $85$ and $87$

a. Explain one disadvantage of using $伪=0.10$ instead of $伪=0.05$ when

performing the test.

b. Explain one disadvantage of taking a random sample of $500$ potatoes instead of $250$ potatoes from the shipment.

No homework? Refer to Exercise 1. The math teachers inspect the

homework assignments from a random sample of 50 students at the school. Only 68% of the students completed their math homework. A significance test yields a P-value of 0.1265.

a. Explain what it would mean for the null hypothesis to be true in this setting.

b. Interpret the P-value.

Restaurant power problems Refer to Exercises $86$ and $88$

a. Explain one disadvantage of using $伪=0.10$ instead of $伪=0.05$ when

performing the test.

b. Explain one disadvantage of taking a random sample of $50$ people instead of $30$ people.

A company that manufactures classroom chairs for high school students

claims that the mean breaking strength of the chairs is 300 pounds. One of the chairs collapsed beneath a 220-pound student last week. You suspect that the manufacturer is exaggerating the breaking strength of the chairs, so you would like to perform a test of $$\begin{gathered} H_0:渭=300 \\ {H}_a:渭<300 \end{gathered}$$where 渭 is the true mean breaking strength of this company鈥檚 classroom chairs.

a. The power of the test to detect that 渭=294 based on a random sample of 30

chairs and a significance level of 伪=0.05 is 0.71. Interpret this value.

b. Find the probability of a Type I error and the probability of a Type II error for the test in part (a).

c. Describe two ways to increase the power of the test in part (a).

Better parking A local high school makes a change that should improve student satisfaction with the parking situation. Before the change, $37\%$ of the school鈥檚 students approved of the parking that was provided. After the change, the principal surveys an SRS of students at the school. She would like to perform a test of $$\begin{gathered} H_0:p=0.37 \\ {H}_a:p>0.37 \end{gathered}$$where p is the true proportion of students at school who are satisfied with the parking

situation after the change.

a. The power of the test to detect that $p=0.45$ based on a random sample of $200$ students and a significance level of $伪=0.05$ is $0.75$ Interpret this value.

b. Find the probability of a Type I error and the probability of a Type II error for the test in part (a).

c. Describe two ways to increase the power of the test in part (a).