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.

Consider the hypothesis test $H_0: μ_1 = μ_2$ against $H_1: μ_1 \gt μ_2$ with known variances $σ_1 = 10$ and $σ_2 = 5$. Suppose that sample sizes $n_1 = 10$ and $n_2 = 15$ and that $\overline{x}_1=24.5$ and $\overline{x}_2=21.3$. Use $\alpha = 0.01$.

a. Test the hypothesis and find the $P$-value.

b. Explain how the test could be conducted with a confidence interval.

c. What is the power of the test in part(a) if $\mu_1$ is 2 units greater than $\mu_2$? Assume that $\alpha=0.05$.

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 etch solution. The observed etch rates are as follows (in mils/min):

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

(a) Do the data support the claim that the mean etch rate is the same for both solutions? In reaching your conclusions, use a fixed-level test with $\alpha = 0.05$ and assume that both population variances are equal. (b) Calculate a P-value for the test in part (a). (c) Find a 95% CI 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 surface finish of 10 metal parts produced in a grinding process is as follows: (in microinches): 10.32, 9.68, 9.92, 10.10, 10.20, 9.87, 10.14, 9.74, 9.80, 10.26. Do the data support the claim that the median value of surface finish is 10 microinches? Use the sign test with $\alpha=0.05 .$ What is the P-value for this test?

For the model of Exercise 12.50(a), test the hypothesis H0: \beta_{4}=0, H1: \beta_{4} \neq 0. Use a P-value in your conclusion.

Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The variance of road octane number for formulation 1 is $σ^2_1 = 1.5$, and for formulation, 2 it is $σ^2_2 = 1.2$. Two random samples of size $n_1 = 15$ and $n_2 = 20$ are tested, and the mean road octane numbers observed are $\overline{x}_1=89.6$ and $\overline{x}_2=92.50$. Assume normality.

a. If formulation 2 produces a higher road octane number than formulation 1, the manufacturer would like to detect it. Formulate and test an appropriate hypothesis using $α = 0.05$. What is the $P$-value?

b. Explain how the question in part (a) could be answered with a 95% confidence interval on the difference in mean road octane number.

c. What sample size would be required in each population if you wanted to be 95% confident that the error in estimating the difference in mean road octane number is less than 1?

Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The variance of road octane number for formulation 1 is $\sigma_{1}^{2}=1.5$ and for formulation 2 it is $\sigma_{2}^{2}=1.2.$ Two random samples of size $n_1 = 15$ and $n_2 = 20$ are tested, and the mean road octane numbers observed are $\overline{x}_{1}=88.85 \text { and } \overline{x}_{2}=92.54.$ Assume normality. (a) Construct a 95% two-sided CI on the difference in mean road octane number. (b) If formulation 2 produces a higher road octane number than formulation 1, the manufacturer would like to detect it. Formulate and test an appropriate hypothesis using the P-value approach.

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: Catalyst 1:  57.9, 66.2, 65.4, 65.4, 65.2, 62.6, 67.6, 63.7, 67.2, 71.0 Catalyst 2:  66.4, 71.7, 70.3, 69.3, 64.8, 69.6, 68.6, 69.4, 65.3, 68.8. a. Find a 95% confidence interval on the difference in mean active concentrations for the two catalysts. Find the P-value. 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). c. Suppose that the true mean difference in active concentration is 5 grams per liter. What is the power of the test to detect this difference if α = 0.05? d. If this difference of 5 grams per liter is really important, do you consider the sample sizes used by the experimenter to be adequate? Does the assumption of normality seem reasonable for both samples?

Consider the hypothesis test H$_0$: μ$_1$ = μ$_2$ against H$_1$: μ$_1$ = μ$_2$. Suppose that sample sizes n$_1$ = 15 and n$_2$ = 15, that $\overline{x}_1$ = 6.2 and $\overline{x}_2$ = 7.8, and that s$^2$ $_1$ = 4 and s$^2$ $_2$ = 6.25. Assume that σ$^2$ $_1$ = σ$^2$ $_2$ and that the data are drawn from normal distributions. Use α = 0.05.
a. Test the hypothesis and find the P-value.
b. Explain how the test could be conducted with a confidence interval.
c. What is the power of the test in part (a) if μ$_1$ is 3 units less than μ$_2$?
d. Assume that sample sizes are equal. What sample size should be used to obtain ß = 0.05 if μ$_1$ is 2.5 units less than μ$_2$? Assume that α = 0.05.

Consider the hypothesis test $H_0: μ_1=μ_2$ against $H_1: μ_1 ≠ μ_2$.

Suppose that sample sizes are $n_1 = 15$ and $n_2 = 15$, that $\overline{x}_1=4.7$ and $\overline{x}_2=7.8$, and that $s_1^2 = 4$ and $s_2^2 = 6.25$.

Assume that $σ_1^2 = σ_2^2$ and that the data are drawn from normal distributions. Use $α = 0.05$.

a) Test the hypothesis and find the $P$-value.

b) Explain how the test could be conducted with a confidence interval.

c) What is the power of the test in part (a) for a true difference in means of $3?$

d) Assume that sample sizes are equal. What sample size should be used to obtain $ß = 0.05$ if the true difference in means is $-2?$ Assume that $α = 0.05$.

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 °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 1 pipe exceeds that of type 2? In reaching your conclusions, use α = 0.05. Calculate a P-value. c. If the mean defl ection temperature for type 1 pipe exceeds that of type 2 by as much as 5°F, it is important to detect this difference with probability at least 0.90. Is the choice of

$$n_1 = n_2 = 15$$

adequate? Use α = 0.05.

Suppose that a quality characteristic is normally distributed with specifications from 20 to 32 units. a. What value is needed for σ to achieve a PCR of 1.5? b. What value for the process mean minimizes the fraction defective? Does this choice for the mean depend on the value of σ?

Find the limit of each sequence. $\left\{\left(1- \frac{4}{n}\right)^{n}\right\}$