A bearing used in an automotive application is supposed to have a nominal inside diameter of 1.5 inches. A random sample of 25 bearings is selected, and the average inside diameter of these bearings is 1.4975 inches. Bearing diameter is known to be normally distributed with standard deviation σ = 0.01 inch. a. Test the hypothesis

$$H_0: μ = 1.5$$

versus

$$H_1: μ ≠ 1.5$$

using α = 0.01. b. What is the P-value for the test in part (a)? Compute the power of the test if the true mean diameter is 1.495 inches. d. What sample size would be required to detect a true mean diameter as low as 1.495 inches if you wanted the power of the test to be at least 0.9?

The proportion of residents in Phoenix favoring the building of toll roads to complete the freeway system is believed to be p = 0.3. If a random sample of 10 residents shows that 1 or fewer favor this proposal, we will conclude that p < 0.3. a. Find the probability of type I error if the true proportion is p = 0.3. b. Find the probability of committing a type II error with this procedure if p = 0.2. c. What is the power of this procedure if the true proportion is p = 0.2?

Supercavitation is a propulsion technology for undersea vehicles that can greatly increase their speed. It occurs above approximately 50 meters per second when the pressure drops sufficiently to allow the water to dissociate into water vapor, forming a gas bubble behind the vehicle. When the gas bubble completely encloses the vehicle, supercavitation is said to occur. Eight tests were conducted on a scale model of an undersea vehicle in a towing basin with the average observed speed $\overline{x}$ = 102.2 meters per second. Assume that speed is normally distributed with a known standard deviation σ = 4 meters per second.
a. Test the hypothesis H$_0$: μ = 100 versus H$_1$: μ < 100 using α = 0.05.
b. What is the P-value for the test in part (a)?
c. Compute the power of the test if the true mean speed is as low as 95 meters per second.
d. What sample size would be required to detect a true mean speed as low as 95 meters per second if you wanted the power of the test to be at least 0.85?
e. Explain how the question in part (a) could be answered by constructing one-sided confidence bound on the mean speed.

Two types of plastic are suitable for use by an electronics component manufacturer. The breaking strength of this plastic is important. It is known that $\sigma_1=\sigma_2=1.0 \mathrm{psi}.$ From a random sample of size $n_1 = 10$ and $n_2 = 12,$ we obtain $\overline{x}_{1}=162.7 \text { and } \overline{x}_{2}=155.4.$ The company will not adopt plastic 1 unless its mean breaking strength exceeds that of plastic 2 by at least 10 psi. Based on the sample information, should it use plastic 1? Use the P-value approach in reaching a decision.

A consumer products company is formulating a new shampoo and is interested in foam height (in millimeters). Foam height is approximately normally distributed and has a standard deviation of 20 millimeters. The company wishes to test H$_0$ : μ = 175 millimeters versus H$_1$: μ > 175 millimeters, using the results of n = 10 samples. Find the type I error probability α if the critical region is $\overline{x}$ > 185. b. What is the probability of type II error if the true mean foam height is 185 millimeters? c. Find ß for the true mean of 195 millimeters.

Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient's body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation $\sigma=0.2$ hour. (a) Is there evidence to support the claim that mean battery life exceeds 4 hours? Use $\alpha=0.05$. (b) Compute the power of the test if the true mean battery life is 4.5 hours. (c) What sample size would be required to detect a true mean battery life of 4.5 hours if we wanted the power of the test to be at least 0.9? (d) Explain how the question in part (a) could be answered by constructing a one-sided confidence bound on the mean life.

A 6-mm-diameter pin is used at connection $C$ of the pedal shown. Knowing that $P=500 \mathrm{~N}$, determine $(a)$ the average shearing stress in the pin, (b) the nominal bearing stress in the pedal at $C,(c)$ the nominal bearing stress in each support bracket at $C$.

A company manufactures ball bearings that are supplied to other companies. The machine that is used to manufacture these ball bearings produces them with a variance of diameters of 025 square millimeter or less. The quality control officer regularly collects a sample of these ball bearings to determine whether or not their variance is within $.025$ square millimeter utilizing confidence intervals and hypothesis testing. If not, the device is halted and modified. Recent random sampling of 23 ball bearings revealed a $.034$ square millimeter variance in the diameters. a. Can you determine that the machine needs to be adjusted using a $5 \%$ significance level? Assume that all ball bearings' diameters are distributed normally. b. Create an interval of 95% confidence for the population variance.

Test the hypothesis that the average content of containers of a particular lubricant is 10 liters if the contents of a random sample of 10 containers are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and 9.8 liters. Use a 0.01 level of significance and assume that the distribution of contents is normal.

If the standard deviation of hole diameter exceeds 0.01 millimeters, there is an unacceptably high probability that the rivet will not fit. Suppose that n = 15 and s = 0.008 millimeter. a. Is there strong evidence to indicate that the standard deviation of hole diameter exceeds 0.01 millimeter? Use α = 0.01. State any necessary assumptions about the underlying distribution of the data. Find the P-value for this test. b. Suppose that the actual standard deviation of hole diameter exceeds the hypothesized value by 50%. What is the probability that this difference will be detected by the test described in part (a)? c. If σ is really as large as 0.0125 millimeters, what sample size will be required to detect this with power of at least 0.8?

The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100, and the standard deviation is 2. You wish to test H$_0$: μ = 100 versus H$_1$: μ ≠ 100 with a sample of n = 9 specimens.
a. If the acceptance region is defined as 98.5 ≤ $\overline{x}$ ≤ 101.5, find the type I error probability α.
b. Find β for the case in which the true mean heat evolved is 103.
c. Find β for the case where the true mean heat evolved is 105. This value of β is smaller than the one found in part (b). Why?

The burning rates of two different solid-fuel propellants used in aircrew escape systems are being studied. It is known that both propellants have approximately the same standard deviation of burning rate; that is, $\sigma_1=\sigma_2=3 \mathrm{cm} / \mathrm{s}.$ Two random samples of $n_1 = 20$ and $n_2 = 20$ specimens are tested; the sample mean burning rates are $\overline{x}_{1}=18.02 \mathrm{cm} / \mathrm{s}$ and $\overline{x}_{2}=24.37 \mathrm{cm} / \mathrm{s}$ (a) Test the hypothesis that both propellants have the same mean burning rate. Use a fixed-level test with $\alpha = 0.05.$ (b) What is the P-value of the test in part (a)? (c) What is the $\beta-error$ of the test in part (a) if the true difference in mean burning rate is 2.5 cm/s? (d) Construct a 95% CI on the difference in means $\mu_1-\mu_2.$ What is the practical meaning of this interval?

An article in the ASCE Journal of Energy Engineering (Vol. 125, 1999) describes a study of the thermal inertia properties of autoclaved aerated concrete used as a building material. Five samples of the material were tested in a structure, and the average interior temperature (0C) reported was as follows: 23.01, 22.22, 22.04, 22.62, and 22.59. (a) Test the hypotheses $H_{0} : \mu=22.5 \text { versus } H_{1} : \mu \neq 22.5$ using $\alpha = 0.05.$ Use the P-value approach. (b) Check the assumption that interior temperature is normally distributed. (c) Find a 95% CI on the mean interior temperature. (d) What sample size would be required to detect a true mean interior temperature as high as 22.75 if we wanted the power of the test to be a least 0.9? Use the sample standard deviation s as an estimate of $\sigma.$

The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 13 defectives. a. Calculate a 95% two-sided CI on the fraction of defective circuits produced by this particular tool. b. Calculate a 95% upper confidence bound on the fraction of defective circuits.