Consider the data on $x=$ compressive strength and $y=$ intrinsic permeability of concrete.

Test for significance of regression using $\alpha=0.05$. Find the $P$-value for this test. Can you conclude that the model specifies a useful linear relationship between these two variables?

Compute the power of the test if the true mean rainfall is 27 acre-feet.

Estimate the standard errors of $\hat{\beta}_1$ and $\hat{\beta}_0$.

In control replication, cells are replicated over a period of two days. Two control mechanisms have been identified-one positive and one negative. Suppose that a replication is observed in three cells. Let $A$ denote the event that all cells are identified as positive, and let $B$ denote the event that all cells are negative. Describe the sample space graphically and display each of the following events:

$$B$$

Pulsed laser deposition technique is a thin film deposition technique with a high-powered laser beam. Twenty-five films were deposited through this technique. The thicknesses of the films obtained are shown in the following table.

$$\begin{array}{cccccc}\text { Film } & \begin{array}{c}\text { Thickness } \\ (\mathbf{n m})\end{array} & \text { Film } & \begin{array}{c}\text { Thickness } \\ (\mathbf{n m})\end{array} & \text { Film } & \begin{array}{c}\text { Thickness } \\ (\mathbf{n m})\end{array} \\ 1 & 28 & 10 & 35 & 19 & 56 \\ 2 & 45 & 11 & 47 & 20 & 49 \\ 3 & 34 & 12 & 50 & 21 & 21 \\ 4 & 29 & 13 & 32 & 22 & 62 \\ 5 & 37 & 14 & 40 & 23 & 34 \\ 6 & 52 & 15 & 46 & 24 & 31 \\ 7 & 29 & 16 & 59 & 25 & 98 \\ 8 & 51 & 17 & 20 & & \\ 9 & 23 & 18 & 33 & & \end{array}$$

Using all the data, compute trial control limits for individual observations and moving-range charts. Determine whether the process is in statistical control. If not, assume that assignable causes can be found to eliminate these samples and revise the control limits.

Test the hypothesis that $\beta_0=0$.

Consider the data on y= chloride concentration in surface streams and x= roadway area.

Test the hypothesis $H_0: \beta_1=0$ versus $H_1: \beta_1 \neq 0$ using the analysis of variance procedure with $\alpha=0.01$.

Consider the situation in the next exercise. After collecting a sample, we are interested in testing $H_0: p=0.10$ versus $H_1: p \neq 0.10$ with $\alpha=0.05$. For the following situation, compute the $p$-value for this test:

$n=500, \widehat{p}=0.095$

Suppose that a quality characteristic is normally distributed with specifications at $120 \pm 20$. The process standard deviation is 6.5 .

Suppose that the process mean is 120. What are the natural tolerance limits? What is the fraction defective? Calculate $P C R$ and $P C R_k$ and interpret these ratios.

Test $H_0: \beta_0=0$ versus $H_1: \beta_0 \neq 0$ using $\alpha=0.05$. What is the $P$-value for this test?

Find the $P$-value for the test in part (a).

Two envelopes, each containing a check, are placed in front of you. You are to choose one of the envelopes, open it, and see the amount of the check. At this point, either you can accept that amount or you can exchange it for the check in the unopened envelope. What should you do? Is it possible to devise a strategy that does better than just accepting the first envelope? Let A and B, A<B, denote the (unknown) amounts of the checks, and note that the strategy that randomly selects an envelope and always accepts its check has an expected return of (A+B) / 2. Consider the following strategy: Let $F(\cdot)$ be any strictly increasing (that is, continuous) distribution function. Choose an envelope randomly and open it. If the discovered check has the value x, then accept it with probability F(x) and exchange it with probability 1-F(x).

a. Show that if you employ the latter strategy, then your expected return is greater than (A+B) / 2.

b. Hint: Condition on whether the first envelope has the value A or B.

c. Now consider the strategy that fixes a value x and then accepts the first check if its value is greater than x and exchanges it otherwise.

a. Show that for any x, the expected return under the x -strategy is always at least (A+B) / 2 and that it is strictly larger than (A+B) / 2 if x lies between A and B.

b. Let X be a continuous random variable on the whole line, and consider the following strategy: Generate the value of X, and if X=x, then employ the x-strategy of part (b). Show that the expected return under this strategy is greater than (A+B) / 2.

Using the regression , Find a $95 \%$ prediction interval for the percent body fat for a man with a BMI of 25.

Consider the data on y= deflection and x= stress level.

Test for significance of regression using $\alpha=0.01$. What is the $P$-value for this test? State the conclusions that result from this test.

Consider the situation in the next exercise. After collecting a sample, we are interested in testing $H_0: p=0.10$ versus $H_1: p \neq 0.10$ with $\alpha=0.05$. For the following situation, compute the $p$-value for this test:

$$n=100, \hat{p}=0.095$$