Recall the sugar content of the syrup in canned peaches. Suppose that the variance is thought to be $\sigma^{2}=18$(milligrams)$^{2}$. A random sample of n = 10 cans yields a sample standard deviation of s = 4.8 milligrams. (a) Test the hypothesis $H_{0}: \sigma^{2}=18$ versus $H_{1}: \sigma^{2} \neq 18$ using $\alpha=0.05$. (b) What is the P-value for this test? (c) Discuss how part (a) could be answered by constructing a 95% two-sided confidence interval for $\sigma$.

Trace the shuffle algorithm for the input 5, 51, 2, 44, 96. Assume that the values of rand are rand(1,5)=1, rand(2,5)=3, rand(3,5)=5, rand(4,5)=5.

Suppose that Y is a discrete random variable with mean

$$\mu$$

and variance

$$\sigma ^ { 2 }$$

and let

$$W = 2 Y$$

. Recalling that the variance is a measure of spread or dispersion, do you expect the variance of W to be larger than, smaller than, or equal to

$$\sigma ^ { 2 } = V ( Y )$$

? Why?

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 sugar content in milligrams of a syrup used to pack canned fruit is measured for $8$ batches of syrup. The measurements are normally distributed, and the sample standard deviation is $s = 3$. Can you conclude that the population standard deviation is greater than $2$? Use the $\alpha = 0.01$ level of significance.

The magnetic flux through a $10.0-\mathrm{cm}^{2}$ loop is $8.0 \mathrm{T} \cdot \mathrm{cm}^{2}$, and the magnetic field at the location of the loop is uniform. Choose all correct answers. (a) The magnitude of the $\vec{B}$ field must satisfy the condition $B \leq 0.8 \mathrm{T}$ (b) The direction of the $\vec{B}$ field cannot be perpendicular to the surface of the loop. (c) If the direction of $\vec{B}$ field is parallel to the normal to the loop's surface, then B = 0.8 T (d) If B = 1.0 T, then the angle between the $\overrightarrow{\boldsymbol{B}}$ field and the normal to the loop's surface is greater than $0 ^ {\circ}$

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?

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

The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the legendary champion, Byron Nelson. Ten randomly selected balls of two different brands are tested and the overall distance measured. The data follow:

$$\begin{array}{l} \text { Brand } 1: 275,286,287,271,283,271,279,275,263,267 \\ \text { Brand } 2: 258,244,260,265,273,281,271,270,263,268 \end{array}$$

(a) Is there evidence that overall distance is approximately normally distributed? Is an assumption of equal variances justified? (b) Test the hypothesis that both brands of ball have equal mean overall distance. Use $\alpha$ = 0.05. (c) What is the P-value of the test statistic in part (b)? (d) What is the power of the statistical test in part (b) to detect a true difference in mean overall distance of 5 yards? (e) What sample size would be required to detect a true difference in mean overall distance of 3 yards with power of approximately 0.75? (f) Construct a 95% two-sided CI on the mean difference in overall distance between the two brands of golf balls.

A rivet is to be inserted into a hole. A random sample of n = 15 parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measurements is s = 0.008 millimeters. Construct a 99% lower confidence bound for σ².

In semiconductor manufacturing, wet chemical etching is often used to remove silicon from the backs of wafers prior to metalization. The etch rate is an important characteristic in this process and known to follow a normal distribution. Two different etching solutions have been compared, using two random samples of 10 wafers for each solution. The observed etch rates are as follows (in mils per minute):

$$\begin{array}{cccc} \hline {\text { Solution 1 }} & & {\text { Solution 2 }} \\ \hline 9.9 & 10.6 & 10.2 & 10.0 \\ 9.4 & 10.3 & 10.6 & 10.2 \\ 9.3 & 10.0 & 10.7 & 10.7 \\ 9.6 & 10.3 & 10.4 & 10.4 \\ 10.2 & 10.1 & 10.5 & 10.3 \\ \hline \end{array}$$

(a) Do the data support the claim that the mean etch rate is the same for both solutions? In reaching your conclusions, use $\alpha=0.05$ and assume that both population variances are equal. (b) Calculate a P-value for the test in part (a). (c) Find a 95% confidence interval on the difference in mean etch rates. (d) Construct normal probability plots for the two samples. Do these plots provide support for the assumptions of normality and equal variances? Write a practical interpretation for these plots.

The deflection temperature under load for two different types of plastic pipe is being investigated. Two random samples of 15 pipe specimens are tested, and the deflection temperatures observed are as follows (in $^{\circ} \mathrm{F}$): Type 1: 206, 188, 205, 187, 194, 193, 207, 185, 189, 213, 192, 210, 194, 178, 205. Type 2: 177, 197, 206, 201, 180, 176, 185, 200, 197, 192, 198, 188, 189, 203, 192. (a) Construct box plots and normal probability plots for the two samples. Do these plots provide support of the assumptions of normality and equal variances? Write a practical interpretation for these plots. (b) Do the data support the claim that the deflection temperature under load for type 2 pipe exceeds that of type 1? In reaching your conclusions, use $\alpha=0.05$. (c) Calculate a P-value for the test in part (b). (d) Suppose that if the mean deflection temperature for type 2 pipe exceeds that of type 1 by as much as $5^{\circ} \mathrm{F}$, it is important to detect this difference with probability at least 0.90. Is the choice of $n_{1}=n_{2}=15$ in part (a) of this problem adequate?

The concentration of active ingredient in a liquid laundry detergent is thought to be affected by the type of catalyst used in the process. The standard deviation of active concentration is known to be 3 grams per liter, regardless of the catalyst type. Ten observations on concentration are taken with each catalyst, and the data follow:

$$\begin{array}{l} \text { Catalyst } 1: 57.9,66.2,65.4,65.4,65.2,62.6,67.6, \\ \qquad 63.7, 67.2,71.0 \\ \text { Catalyst } 2: 66.4,71.7,70.3,69.3,64.8,69.6,68. \\ \qquad 69.4, 65.3, 68.8 \end{array}$$

(a) Find a 95% confidence interval on the difference in mean active concentrations for the two catalysts. (b) Is there any evidence to indicate that the mean active concentrations depend on the choice of catalyst? Base your answer on the results of part (a).