(a) state the null and alternative hypotheses and identify which represents the claim, (b) describe type I and type II errors for a hypothesis test of the claim, (c) explain whether the hypothesis test is left-tailed, right-tailed, or two-tailed, (d) explain how you should interpret a decision that rejects the null hypothesis, and (e) explain how you should interpret a decision that fails to reject the null hypothesis. A polling organization reports that the proportion of U.S. adults who have volunteered their time or donated money to help clean up the environment is 65%.

Brower Co. is considering the following alternative financing plans:

$$\begin{array}{lcc} & \textbf { Plan 1 } & \textbf { Plan 2 } \\ \hline \text { Issue } 10 \% \text { bonds (at face value) } & \$ 4,000,000 & \$ 2,500,000 \\ \text { Issue preferred } \$ 2.50 \text { stock, } \$ 25 \text { par } & - & 3,000,000 \\ \text { Issue common stock, } \$ 10 \text { par } & 4,000,000 & 2,500,000 \end{array}$$

Income tax is estimated at 40% of income. Determine the earnings per share of common stock, assuming income before bond interest and income tax is $2,000,000.

Use the applet Hypotheses Test for a Proportion to investigate the relationships between the probabilities of Type I and Type II errors occurring at levels $.05$ and $.01$. For this exercise, use $n=100$, true $p=0.5$, and alternative not equal. Set null $p=.5$. What happens to the proportion of times the null hypothesis is rejected at the $.05$ level and at the $.01$ level as the applet is run more and more times? What type of error has occurred when the null hypothesis is rejected in this situation? Based on your results, is this type of error more likely to occur at level $.05$ or at level .01? Explain.

A pharmaceutical manufacturer is concerned that the impurity concentration in pills does not exceed 3%. It is known that from a particular production run, impurity concentrations follow a normal distribution with standard deviation 0.4%. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be 3.07%.

a. Test, at the 5% level, the null hypothesis that the population mean impurity concentration is 3% against the alternative that it is more than 3%.

b. Find the probability of a 5%-level test rejecting the null hypothesis when the true mean impurity concentration is 3.10%.

A chocolatier claims that his candy averages at least 81 pieces per bag. Conrad randomly selects bags of the candies and counts the pieces. Assume that $\sigma=1.2.$

$$\text{Number of Pieces} \\ \begin{array}{|l|l|l|l|l|l|l|l|l|} \hline 81 & 80 & 82 & 82 & 83 & 82 & 84 & 81 & 81 \\ \hline 80 & 83 & 83 & 82 & 81 & 80 & 84 & 81 & 81 \\ \hline \end{array}$$

Write the null and alternative hypotheses, and state which hypothesis represents the claim.

Fly Co. is considering the following alternative financing plans:

$$\begin{array}{lrr} & \textbf { Plan 1 } & \textbf { Plan 2 } \\ \hline \text { Issue } 12 \% \text { bonds (at face value) } & \$ 10,000,000 & \$ 5,000,000 \\ \text { Issue preferred } \$ 1.75 \text { stock, } \$ 20 \text { par } & - & 8,000,000 \\ \text { Issue common stock, } \$ 20 \text { par } & 10,000,000 & 7,000,000 \end{array}$$

Income tax is estimated at 40% of income. Determine the earnings per share of common stock, assuming income before bond interest and income tax is $2,000,000.

Frey Co. is considering the following alternative financing plans:

$$\begin{array}{lrr} & \text { Plan 1 } & \text { Plan 2 } \\ \hline \text { Issue } 5 \% \text { bonds (at face value) } & \$ 6,000,000 & \$ 2,000,000 \\ \text { Issue preferred } \$ 1 \text { stock, } \$ 20 \text { par } & - & 6,000,000 \\ \text { Issue common stock, } \$ 25 \text { par } & 6,000,000 & 4,000,000 \end{array}$$

Income tax is estimated at 40% of income.

Determine the earnings per share of common stock, assuming that income before bond interest and income tax is $800,000.

Based on a sample of 30 observations, the population regression model

$$y_i=\beta_0+\beta_1 x_i+\varepsilon_i$$

was estimated. The least squares estimates obtained were as follows:

$$b_0=10.1 \text { and } b_1=8.4$$

The regression and error sums of squares were as follows:

$$S S R=128 \text { and } \quad S S E=286$$

a. Find and interpret the coefficient of determination.

b. Test at the $10 \%$ significance level against a twosided alternative the null hypothesis that $\beta_1$ is 0 .

c. Find

$$\sum_{i=1}^{30}\left(x_i-\bar{x}\right)^2$$

In Relief from Arthritis published by Thorsons Publishers, Ltd., John E. Croft claims that over 40% of those who suffer from osteoarthritis receive measurable relief from an ingredient produced by a particular species of mussel found ff the coast of New Zealand. To test this claim, the mussel extract is to be given to a group of 7 osteoarthritic patients. If 3 or more of the patients receive relief, we shall not reject the null hypothesis that p =0 .4; otherwise, we conclude that p < 0.4. (a) Evaluate $\alpha$ assuming that p = 0.4. (b) Evaluate $\beta$ for the alternative p = 0.3.

Consider the following alternatives:

$$\begin{matrix} \text{ } & \text{A} & \text{B} & \text{C}\\ \text{Initial cost} & \text{\$10,000} & \text{\$15,000} & \text{\$20,000}\\ \text{Uniform annual benefit} & \text{1000} & \text{1762} & \text{5548}\\ \text{Useful life (years)} & \text{Infinite} & \text{20} & \text{5}\\ \end{matrix}$$

Alternatives B and C are replaced at the end of their useful lives with identical replacements. Use an infinite analysis period. (a) Construct a choice table for interest rates from 0% to 100%. (b) At an 8% interest rate, which alternative is better?

Xanadu Mining is considering three mutually exclusive alternatives as shown in the table below. $MARR$ is 10 percent/year. Based on an internal rate of return analysis, which alternative should be recommended?

$$\begin{array}{|c|c|c|c|c|c|} \hline \text { EOY } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { A001 } & -\$ 210 & \$ 80 & \$ 90 & \$ 100 & \$ 110 \\ \hline \text { B002 } & -\$ 110 & \$ 60 & \$ 60 & \$ 60 & \$ 70 \\ \hline \text { C003 } & -\$ 160 & \$ 80 & \$ 80 & \$ 80 & \$ 80 \\ \hline \end{array}$$

The consultant who advised the city on building the parking garage described predicted that the parking revenues would average $130 per day. Do the data from these two months suggest the consultant's estimate may have been incorrect? a) Write the null hypothesis. b) Decide whether you should conduct a one- or two-tail test and write an appropriate alternative hypothesis. c) Confirm that the assumptions and conditions for inference based on t-models are satisfied. d) Find the value of t and the P-value based on these data. e) State your conclusion.

A random sample of 50 students was asked what salary the college should be prepared to pay to attract the right individual to coach the football team. An independent random sample of 50 faculty members was asked the same question. The 100 salary figures were then pooled and ranked in order (with rank 1 assigned to the lowest salary). The sum of the ranks for faculty members was 2,024. Test the null hypothesis that there is no difference between the central locations of the distributions of salary proposals of students and faculty members against the alternative that in the aggregate students would propose a higher salary to attract a football coach.

Javier is an IE at Lobos Manufacturing. He has been studying process line G to determine if an automated system would be preferred to the existing labor intensive system. If Lobos wants to earn at least 20% and uses a 15-year planning horizon, which alternative is preferred.