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).

The simple beam AB supports two equal concentrated loads P, one acting downward and the other upward.

Determine the angle of rotation $\theta_A$ at the left-hand end, the deflection $\delta_1$ under the downward load, and the deflection $\delta_2$ at the midpoint of the beam.

Consider the situation described. What sample size would be required in each population if we wanted the error in estimating the difference in mean burning rates to be less than 4 cm/s with 99% confidence? 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 $\overline{x}_{1}=18.02 cm/s$ and $\overline{x}_{2}=24.37$(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 β-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?

A researcher claims that at least 10% of all football helmets have manufacturing flaws that could potentially cause injury to the wearer. A sample of 200 helmets revealed that 16 helmets contained such defects. (a) Does this finding support the researcher's claim? Use $\alpha=0.01$. (b) Find the P-value for this test.

Mathematician tosses coin 10,000 times! The South African mathematician John Kerrich, while a prisoner of war during World War II, tossed a coin 10,000 times and obtained 5067 heads.

(a) Is this significant evidence at the $5 \%$ level that the probability that Kerrich's coin comes up heads is not 0.5 ?

(b) Give a 95% confidence interval to see what probabilities of heads are roughly consistent with Kerrich's result.

Refer to the earlier exercise.

What would happen to the sampling distribution of the sample minimum if the sample size were. n=5 instead? Explain your answer.

A research engineer for a tire manufacturer is investigating tire life for a new rubber compound. She has built 10 tires and tested them to end-of-life in a road test. The sample mean and standard deviation are 61,492 and 3035 kilometers, respectively. (a) The engineer would like to demonstrate that the mean life of this new tire is in excess of 60,000 km. Formulate and test appropriate hypotheses, being sure to state (test, if possible) assumptions, and draw conclusions using the P-value approach. (b) Suppose that if the mean life is as long as 61,000 km, the engineer would like to detect this difference with probability at least 0.90. Was the sample size n = 10 used in part (a) adequate? Use the sample standard deviation s as an estimate of σ in reaching your decision. (c) Find a 95% one-sided lower confidence bound on mean tire life. (d) Use the bound found in part (c) to test the hypothesis.

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 reported here (in 0F):

$$\begin{matrix} \quad & \text{Type 1} & \quad & \quad & \quad & \text{Type 2}\\ \text{206} & \text{193} & \text{192} & \quad & \text{177} & \text{176} & \text{198}\\ \text{188} & \text{207} & \text{210} & \quad & \text{197} & \text{185} & \text{188}\\ \text{205} & \text{185} & \text{194} & \quad & \text{206} & \text{200} & \text{189}\\ \text{187} & \text{189} & \text{178} & \quad & \text{201} & \text{197} & \text{203}\\ \text{194} & \text{213} & \text{205} & \quad & \text{180} & \text{192} & \text{192}\\ \end{matrix}$$

(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 a P-value in answering this question. (c) Suppose that if the mean deflection temperature for type 2 pipe exceeds that of type 1 by as much as 50F, it is important to detect this difference with probability at least 0.90 when $\alpha = 0.05.$ Is the choice of $n_1 = n_2 = 15$ in part (b) of this problem adequate?

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?

While he was a prisoner of war during World War II, John Kerrich tossed a coin 10,000 times. He got 5067 heads. If the coin is perfectly balanced, the probability of a head is 0.5. Is there reason to think that Kerrich's coin was not balanced? To answer this question, use a Normal distribution to estimate the probability that tossing a balanced coin 10,000 times would give a count of heads at least this far from 5000 (that is, at least 5067 heads or at most 4933 heads.)

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?

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?

A bearing used in an automotive application is suppose 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 $\sigma=0.01$ inch. (a) Test the hypotheses $H_{0}: \mu=1.5$ versus $H_{1}: \mu \neq 1.5$ using $\alpha=0.01$. (b) Compute the power of the test if the true mean diameter is 1.495 inches. (c) What sample size would be required to detect a true mean diameter as low as 1.495 inches 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 two-sided confidence interval on the mean diameter.