The P-value for a two-sided test of the null hypothesis : population mean =15 is 0.03.

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

A blogger claims that U.S. adults drink an average of five 8-ounce glasses (that's 40 ounces ) of water per day. Researchers wonder if this claim is true, so they ask a random sample of 24 U.S. adults about their daily water intake. A graph of the data shows a roughly symmetric shape with no outliers. Do we have convincing evidence that the amount of water U.S. children drink per day differs from 40 ounces? Justify your answer.

Every road has one at some point-construction zones that have much lower speed limits. To see if drivers obey these lower speed limits, a police officer uses a radar gun to measure the speed (in miles per hour, or mph) of a random sample of 10 drivers in a 25 mph construction zone. Here are the data.

$$\begin{array}{llllllllll} 27 & 33 & 32 & 21 & 30 & 30 & 29 & 25 & 27 & 34 \end{array}$$

We want to perform a test at the $\alpha$=0.05 significance level of $H_0: \mu=25$ versus $H_a: \mu>25$, where $\mu=$ the mean speed of all cars that drive through the construction zone.

Explain why the data give some evidence for $H_a$.

People of taste are supposed to prefer fresh-brewed coffee to the instant variety. On the other hand, perhaps many coffee drinkers just want their caffeine fix. A skeptic claims that only half of all coffee drinkers prefer fresh-brewed coffee. To test this claim, we ask a random sample of 50 coffee drinkers in a small city to take part in a study. Each person tastes two unmarked cups - one containing instant coffee and one containing fresh-brewed coffee - and says which he or she prefers. We find that 36 of the 50 choose the fresh coffee. We presented the two cups to each coffee drinker in a random order, so that some people tasted the fresh coffee first, while others drank the instant coffee first. Why do you think we did this?

A student group claims that first-year students at a university must study 2.5 hours per night during the school week. A skeptic suspects that they study less than that on the average. A class survey finds that the average study time claimed by 269 students is

$$\bar{x} = 137 minutes$$

. Regard these students as a random sample of all first-year students and suppose we know that study times follow a Normal distribution with standard deviation 65 minutes. What conclusion would you draw? Give appropriate evidence to support your conclusion.

Statistics students Daniel and Chad decided to investigate whether green labeling makes consumers believe that a product is natural. They took a random sample of 40 people, and served each person two cups of tea-one in a green cup and one in a clear cup. (Unknown to the participants, both cups were filled with the same type of tea.) Daniel and Chad asked each person, "Which cup of tea do you believe has a more natural flavor?" Twenty-nine of the 40 people said the one in the green cup. Based on these data, Daniel and Chad want to carry out a test at the $\alpha$=0.05 significance level of

$$\begin{aligned} & H_0: p=0.50 \\ & H_a: p>0.50 \end{aligned}$$

where p= the true proportion of people who would identify the green cup of tea as having the more natural flavor.

Calculate the standardized test statistic.

Jason reads a report that says 85% of U.S. high school students have a computer at home. He believes the proportion is smaller than 0.85 at his large high school. Jason chooses an SRS of 80 students and finds that 61 have a computer at home. He would like to carry out a test at the $\alpha$=0.05 significance level of

$$\begin{aligned} & H_0: p=0.85 \\ & H_a: p<0.85 \end{aligned}$$

where p= the proportion of all students at Jason's high school who have a computer at home. Check if the conditions for performing the significance test are met.

You are the president of Duke Company and are leading the company through the process of incorporation. The company has determined that common stock shares will be issued, but several key executives at Duke are not quite sure they understand the preemptive right feature associated with common shares. Prepare a memo to your executive team outlining the meaning of this right.

Some addition problems are easier to compute with fractions and some are easier to do with decimals. For example, $\frac 1 7 + \frac 5 7$ is easier to compute than $0.\overline{142857}$ +$0.\overline{714285}$ and 0.4 + 0.25 is easier to compute than $\frac 2 5 + \frac 1 4$ . Describe situations in which you think it would be easier to compute the additions with fractions than with decimals, and vice versa.

The Environmental Protection Agency has determined that safe drinking water should contain no more than 1.3 milligrams per liter (mg/I) of copper, on average. To test water from a new source, you collect water in small bottles at each of 30 randomly selected locations. The mean copper content of your bottles is 1.36 mg/I and the standard deviation is 0.18 mg/I. You perform a test of $H_{0}: \mu=1.3$ versus $H_{a}: \mu>1.3,$ where $\mu$ is the true mean copper content of the water from the new source. The test yields a P-value of 0.0391. What conclusion would you make for each of the following significance levels? $\alpha=0.05$

The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students' attitudes toward school and study habits. Scores range from 0 to 200. Higher scores indicate more positive attitudes. The mean score for U.S. college students is about 115. A teacher suspects that older students have better attitudes toward school. She gives the SSHA to an SRS of 45 of the over 1000 students at her college who are at least 30 years of age. Check the conditions for carrying out a significance test of the teacher's suspicion.

The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students' study habits and attitude toward school. Scores range from 0 to 200. The mean score for U.S. college students is about 115, and the standard deviation is about 30. A teacher suspects that older students have better attitudes toward school. She gives the SSHA to 25 students who are at least 30 years old. Assume that scores in the population of older students are Normally distributed with standard deviation $\sigma=6$. The teacher wants to test the hypotheses

$$\begin{array}{l}{H_{0} : \mu=115} \\ {H_{a} : \mu>115}\end{array}$$

(a) What is the sampling distribution of the mean score $\overline{x}$ of a sample of 25 older students if the null hypothesis is true? Sketch the density curve of this distribution. (b) Suppose that the sample data give $\overline{x}=118.6$. Mark this point on the axis of your sketch. In fact, the outcome was $\overline{x}=125.7$. Mark this point on your sketch. Using your sketch, explain in simple language why one outcome is good evidence that the mean score of all older students is greater than 115 and why the other outcome is not. (c) Shade the area under the curve that is the P-value for the sample result $\overline{x}=118.6$.

The mean weight of loaves of bread produced at the bakery where you work is supposed to be 1 pound. You are the supervisor of quality control at the bakery, and you are concerned that employees are making loaves that are too light. Suppose you weigh an SRS of 50 bread loaves and find that the mean weight is 0.975 pound. State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

The mean weight of loaves of bread produced at a bakery is supposed to be 1 pound. The quality control supervisor at the bakery is concerned that employees are making loaves that are too light. So the supervisor weighs an SRS of 50 bread loaves and finds that the mean weight is 0.975 pound.

The P-value for the test in the first part is 0.0806. Interpret the P-value.