An article in Electronic Components and Technology Conference (Vol. 52, 2001) describes a study comparing single versus dual spindle saw processes for copper metallized wafers. A total of 15 devices of each type were measured for the width of the backside chipouts, $\overline{x}_{\text {single}}=66.385, s_{\text {single}}=7.895 \text { and } \overline{x}_{\text {double}}=45.278, s_{\text {double}}=8.612$ (a) Do the sample data support the claim that both processes have the same chip outputs? Assume that both populations are normally distributed and have the same variance. Answer this question by finding and interpreting the P-value for the test. (b) Construct a 95% two-sided confidence interval on the mean difference in spindle saw process. Compare this interval to the results in part (a). (c) If the $beta-error$ of the test when the true difference in chip outputs is 15 should not exceed 0.1 when $\alpha = 0.05,$ what sample sizes must be used?

The advertised claim for batteries for cell phones is set at 48 operating hours with proper charging procedures. A study of 5000 batteries is carried out and 15 stop operating prior to 48 hours. Do these experimental results support the claim that less than 0.2 percent of the company’s batteries will fail during the advertised time period, with proper charging procedures? Use a hypothesis-testing procedure with α = 0.01.

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.

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 $\sigma_{1}=\sigma_{2}=3$ centimeters per second. Two random samples of $n_{1}=20$ and $n_{2}=20$ specimens are tested; the sample mean burning rates are $\bar{x}_{1}=18$ centimeters per second and $\bar{x}_{2}=24$ centimeters per second. (a) Test the hypothesis that both propellants have the same mean burning rate. Use $\alpha=0.05$. (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 centimeters per second? (d) Construct a 95% confidence interval on the difference in means $\mu_{1}-\mu_{2}$. What is the practical meaning of this interval?

A particle moves along a straight line with a velocity in millimeters per second given by v = 400 – 16t2 , where t is in seconds. Calculate the net displacement $\Delta$s and total distance D traveled during the first 6 seconds of motion.

The hii are often used to denote leverage—that is, a point that is unusual in its location in the x-space and that may be influential. Generally, the ith point is called a leverage point if hii exceeds 2p/n, which is twice the average size of all the hat diagonals. Recall that p = k + 1. (a) The table contains the hat matrix diagonal for the wire bond pull strength data. Find the average size of these elements. (b) Based on the criterion given, are there any observations that are leverage points in the data set? Influence Diagnostics for the Wire Bond Pull Strength Data

$$\begin{matrix} \text{Observations i} & h_a & \text{Cook's Distance Measure} D_i & \text{Observations i} & h_a & \text{Cook's Distance Measure } D_i\\ \text{1} & \text{0.1573} & \text{0.035} & \text{14} & \text{0.1129} & \text{0.003}\\ \text{2} & \text{0.1116} & \text{0.012} & \text{15} & \text{0.0737} & \text{0.187}\\ \text{3} & \text{0.1419} & \text{0.060} & \text{16} & \text{0.0879} & \text{0.001}\\ \text{4} & \text{0.1019} & \text{0.021} & \text{17} & \text{0.2593} & \text{0.565}\\ \text{5} & \text{0.0418} & \text{0.024} & \text{18} & \text{0.2929} & \text{0.155}\\ \text{6} & \text{0.0749} & \text{0.007} & \text{19} & \text{0.0962} & \text{0.018}\\ \text{7} & \text{0.1181} & \text{0.036} & \text{20} & \text{0.1473} & \text{0.000}\\ \text{8} & \text{0.1561} & \text{0.020} & \text{21} & \text{0.1296} & \text{0.052}\\ \text{9} & \text{0.1280} & \text{0.160} & \text{22} & \text{0.1358} & \text{0.028}\\ \text{10} & \text{0.0413} & \text{0.001} & \text{23} & \text{0.1824} & \text{0.002}\\ \text{11} & \text{0.0925} & \text{0.013} & \text{24} & \text{0.1091} & \text{0.040}\\ \text{12} & \text{0.0526} & \text{0.001} & \text{25} & \text{0.0729} & \text{0.000}\\ \text{13} & \text{0.0820} & \text{0.001}\\ \end{matrix}$$

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?

True or False. $\ln e^{x}=x$ for all real numbers.

Consider the fatty acid content of margarine described. (a) Construct a 95% PI for the fatty acid content of a single package of margarine. (b) Find a tolerance interval for the fatty acid content that includes 95% of the margarine packages with 99% confidence. A particular brand of diet margarine was analyzed to determine the level of polyunsaturated fatty acid (in percent). A sample of six packages resulted in the following data: 16.8, 17.2, 17.4, 16.9, 16.5, and 17.1. (a) It is important to determine if the mean is not 17.0. Test an appropriate hypothesis, using the P-value approach. What are your conclusions? Use a normal probability plot to test the normality assumption. (b) Suppose that if the mean polyunsaturated fatty acid content is $\mu=17.5,$ it is important to detect this with probability at least 0.90. Is the sample size n = 6 adequate? Use the sample standard deviation to estimate the population standard deviation $\sigma.$ Use $\alpha = 0.01.$ (c) Find a 99% two-sided CI on the mean µ. Provide a practical interpretation of this interval.

Find the effective annual rate of a 6% annual rate, compounded continuously.

Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst 1, resulting in an average yield of 86 and a sample standard deviation of 3. Fifteen batches were prepared using catalyst 2, and they resulted in an average yield of 89 with a standard deviation of 2. Assume that yield measurements are approximately normally distributed with the same standard deviation. a. Is there evidence to support a claim that catalyst 2 produces a higher mean yield than catalyst 1? Use α = 0.01. b. Find a 99% confidence interval on the difference in mean yields that can be used to test the claim in part (a).

Two catalysts in a batch chemical process, are being compared for their effect on the output of the process reaction. A sample of 12 batches was prepared using catalyst 1, and a sample of 10 batches was prepared using catalyst 2. The 12 batches for which catalyst 1 was used in the reaction gave an average yield of 85 with a sample standard deviation of 4, and the 10 batches for which catalyst 2 was used gave an average yield of 81 and a sample standard deviation of 5. Find a 90% confidence interval for the difference between the population means, assuming that the populations are approximately normally distributed with equal variances.

Consider the hypothesis test $H_0: μ_1 = μ_2$ against $H_1: = μ_1 ≠ μ_2$ with known variances $σ = 10$ and $σ_2 = 5$. Suppose that sample sizes $n_1 = 10$ and $n_2 = 15$ and that $\overline{x_1}=4.7$ and $\overline{x_2}=7.8$. 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 $3$? Assume that $α = 0.05$.

The sugar content of the syrup in canned peaches is normally distributed. A random sample of n = 10 cans yields a sample standard deviation of s = 4.8 milligrams. Calculate a 95% two-sided confidence interval for σ.