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

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

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) 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%.

The following statement appeared on a box of Tide laundry detergent: “Individual packages of Tide may weigh slightly more or less than the marked weight due to normal variations incurred with high speed packaging machines, but each day’s production of Tide will average slightly above the marked weight.” a. Explain in statistical terms what the statement means. b. Describe in words a hypothesis test for checking the statement. c. Suppose that the marked weight is 76 ounces. State in words the null and alternative hypotheses for the hypothesis test. Then express those hypotheses in statistical terminology.

In our discussion of TCP congestion control in earlier Section, we implicitly assumed that the TCP sender always had data to send. Consider now the case that the TCP sender sends a large amount of data and then goes idle (since it has no more data to send) at $t_1$. TCP remains idle for a relatively long period of time and then wants to send more data at $t_2$. What are the advantages and disadvantages of having TCP use the cwnd and ssthresh values from $t_1$ when starting to send data at $t_2$ ? What alternative would you recommend? Why?

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.

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?

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.

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.

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.

A bank conducts a survey in which it randomly samples 400 of its customers. The survey asks the customers which way they use the bank the most:

(1) interacting with a teller at the bank,

(2) using ATMs, or

(3) using the bank's online banking service.

It also asks their level of satisfaction with the service they most often use (on a scale of 0 to 10 with 0 = very poor and 10 = excellent). Does mean satisfaction differ according to how they most use the bank?

Identifying notation, state the null and alternative hypotheses for conducting an ANOVA with data from the survey.

Following are some summaries of the data from two separate samples drawn from normal distributions:

$$\begin{array}{ccc} \hline \text { Sample } & n & s^2 \\ \hline 1 & 11 & 3.5 \\ 2 & 16 & 9.1 \\ \hline \end{array}$$

At the $5$ significance level, you want to test the null hypothesis, which states that the two population standard deviations are equal, against the twosided alternative. (a)Calculate the test statistic in step. (b) Obtain the necessary value from Table $mathrmE$ to do the significance test. (c)What do you deduce from ?