The following data represent the running times of films produced by two motion-picture companies:

$$\begin{matrix} \text{Company} & \text{Time (minutes) }\\ \text{1} & \text{102} & \text{86} & \text{98} & \text{109} & \text{92}\\ \text{2} & \text{81} & \text{165} & \text{97} & \text{134} & \text{92} & \text{87} & \text{114}\\ \end{matrix}$$

Test the hypothesis that the average running time of films produced by company 2 exceeds the average running time of films produced by company 1 by 10 minutes against the one-sided alternative that the difference is less than 10 minutes. Use a 0.1 level of significance and assume the distributions of times to be approximately normal with unequal variances.

A manufacturer claims that the average tensile strength of thread A exceeds the average tensile strength of thread B by at least 12 kilograms. To test this claim, 50 pieces of each type of thread were tested under similar conditions. Type A thread had an average tensile strength of 86.7 kilograms with a standard deviation of 6.28 kilograms, while type B thread had an average tensile strength of 77.8 kilograms with a standard deviation of 5.61 kilograms. Test the manufacturer’s claim using a 0.05 level of significance.

In a study conducted at Virginia Tech on the development of ectomycorrhizal, a symbiotic relationship between the roots of trees and a fungus, in which minerals are transferred from the fungus to the trees and sugars from the trees to the fungus, 20 northern red oak seedlings exposed to the fungus Pisolithus tinctorus were grown in a greenhouse. All seedlings were planted in the same type of soil and received the same amount of sunshine and water. Half received no nitrogen at planting time, to serve as a control, and the other half received 368 ppm of nitrogen in the form $\mathrm{NaNO}_{3}.$ The stem weights, in grams, at the end of 140 days were recorded as follows:

$$\begin{matrix} \text{No Nitrogen} & \text{Nitrogen}\\ \text{0.32} & \text{0.26}\\ \text{0.53} & \text{0.43}\\ \text{0.28} & \text{0.47}\\ \text{0.37} & \text{0.49}\\ \text{0.47} & \text{0.52}\\ \text{0.43} & \text{0.75}\\ \text{0.36} & \text{0.79}\\ \text{0.42} & \text{0.86}\\ \text{0.38} & \text{0.62}\\ \text{0.43} & \text{0.46}\\ \end{matrix}$$

Construct a 95% confidence interval for the difference in the mean stem weight between seedlings that receive no nitrogen and those that receive 368 ppm of nitrogen. Assume the populations to be normally distributed with equal variances.

A marketing expert for a pasta-making company believes that 40% of pasta lovers prefer lasagna. If 9 out of 20 pasta lovers choose lasagna over other pastas, what can be concluded about the expert’s claim? Use a 0.05 level of significance.

In a research report, Richard H. Weindruch of the UCLA Medical School claims that mice with an average life span of 32 months will live to be about 40 months old when 40% of the calories in their diet are replaced by vitamins and protein. Is there any reason to believe that $\mu<40$ if 64 mice that are placed on this diet have an average life of 38 months with a standard deviation of 5.8 months? Use a P-value in your conclusion.

A random sample of $n_{1}=16$ communities in western Kansas gave the following information for people under 25 years of age. $x_{1}:$ Rate of hay fever per 1000 population for people under 25

$$\begin{array}{rccccccc}98 & 90 & 120 & 128 & 92 & 123 & 112 & 93 \\ 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88\end{array}$$

A random sample of $n_{2}=14$ regions in western Kansas gave the following information for people over 50 years old. $x_{2}:$ Rate of hay fever per 1000 population for people over 50

$$\begin{array}{lcccccc}95 & 110 & 101 & 97 & 112 & 88 & 110 \\ 79 & 115 & 100 & 89 & 114 & 85 & 96\end{array}$$

\begin{aligned} i. Use a calculator to verify that $\bar{x}_{1} \approx 109.50,$ $s_{1} \approx 15.41,$ $\bar{x}_{2} \approx 99.36,$ and $s_{2} \approx 11.57$. ii. Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use $\alpha=0.05$. What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic.

The average height of females in the freshman class of a certain college has historically been 162.5 centimeters with a standard deviation of 6.9 centimeters. Is there reason to believe that there has been a change in the average height if a random sample of 50 females in the present freshman class has an average height of 165.2 centimeters? Use a P-value in your conclusion. Assume the standard deviation remains the same.

Execute these steps.

a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results.

Employ the traditional method of hypothesis testing unless otherwise specified.

In a random sample of 200 men, 130 said they used seat belts. In a random sample of 300 women, 63 said they used seat belts. Test the claim that men are more safety-conscious than women, at $\alpha=0.01$. Use the $P$-value method.

In the study Germination and Emeryence of Broccoli, conducted by the Department of Horticulture at Virginia Tech, a researcher found that at $5^\circ C,$ 10 broccoli seeds out of 20 germinated, while at $15^\circ C,$ 15 out of 20 germinated. Compute a 95% confidence interval for the difference between the proportions of germination at the two different temperatures and decide if there is a significant difference.

find the center and radius of the circle. Then sketch the graph of the circle. x^2 + y^2 = 16

Engineers at a large automobile manufacturing company are trying to decide whether to purchase brand A or brand B tires for the company’s new models. To help them arrive at a decision, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are as follows:

$$\begin{aligned} \text { Brand } A : & \overline{x}_{1}=37,900 \text { kilometers, } \\ & s_{1}=5100 \text { kilometers. } \\ \text { Brand } B : & \overline{x}_{1}=39,800 \text { kilometers, } \\ & s_{2}=5900 \text { kilometers. } \end{aligned}$$

Test the hypothesis that there is no difference in the average wear of the two brands of tires. Assume the populations to be approximately normally distributed with equal variances. Use a P-value.

A study was made to determine if the subject matter in a physics course is better understood when a lab constitutes part of the course. Students were randomly selected to participate in either a 3-semesterhour course without labs or a 4-semester-hour course with labs. In the section with labs, 11 students made an average grade of 85 with a standard deviation of 4.7, and in the section without labs, 17 students made an average grade of 79 with a standard deviation of 6.1. Would you say that the laboratory course increases the average grade by as much as 8 points? Use a P-value in your conclusion and assume the populations to be approximately normally distributed with equal variances.

A study was conducted by the Department of Zoology at Virginia Tech to determine if there is a significant difference in the density of organisms at two different stations located on Cedar Run, a secondary stream in the Roanoke River drainage basin. Sewage from a sewage treatment plant and overflow from the Federal Mogul Corporation settling pond enter the stream near its headwaters. The following data give the density measurements, in number of organisms per square meter, at the two collecting stations:

$$\begin{matrix} \text{Number of Organisms per Square Meter }\\ \text{Station 1 } & \quad & \text{Station 2 }\\ \text{5030} & \text{4980} & \text{2800 } & \text{2810}\\ \text{13,700} & \text{11,910 } & \text{ 4670 } & \text{1330}\\ \text{10,730 } & \text{8130} & \text{6890} & \text{3320}\\ \text{11,400 } & \text{26,850 } & \text{7720} & \text{1230}\\ \text{860} & \text{17,660 } & \text{7030} & \text{2130}\\ \text{2200} & \text{22,800 } & \text{7330} & \text{2190}\\ \text{4250} & \text{1130}\\ \text{15,040 } & \text{1690}\\ \end{matrix}$$

Can we conclude, at the 0.05 level of significance, that the average densities at the two stations are equal? Assume that the observations come from normal populations with different variances.