1. Based on a sample of 30 observations, the population regression model y_t = β₀ + β₁x_i + ε_i was estimated. The least squares estimates obtained were as follows:
b_o = 10.1 and b₁ = 8.4.

The regression and error sums of squares were as follows:

SSR= 128 and SSE= 286.

a) Find and interpret the coefficient of determination.

b) Find

c) Test at the 10% significance level against a two sided alternative the null hypothesis that is 0.

2. An insurance company employs agents on a commission basis. It claims that in their first-year agents will earn a mean commission of at least $40,000 and that population standard deviation is no more than $6,000. A random sample of nine agents filed for commission in the first year, 

Where is measured in thousands of dollars and the population distribution can be assumed to be normal. Test at 5% level, the null hypothesis that the population mean is at least $40,000.

3. A wine producer claims that the proportion of its customers who cannot distinguish its product from frozen grape juice is at most 0.09. The producer decides to test this null hypothesis against the alternative that the true proportion is more than 0.09. The decision rule adopted is to reject the null hypothesis if the sample proportion of people who cannot distinguish between these two flavors exceed 0.14.

a) If a random sample of 400 customers is selected, what is the probability of Type I error using this decision rule?

b) Suppose that the true proportion of customers who cannot distinguish between these flavors is 0.20. If a random sample of 100 customers is selected, what is the probability of a type II error?

4. Random samples of employees were drawn in fast food restaurants where the employer provides a training program. Of a sample of 67 employees who had not finished high school, 11 had participated in a training program provided by their current employer. Of an independent random sample of 113 employees who had completed high school but had not attended college, 27 had participated. Test at 1% level, the null hypothesis that the participation rates are the same for the two groups against the alternative that the rate is lower for those who have not completed high school.

5. For a random sample of 526 firms, the sample correlation between the proportion of a firm's officers who are directors and a risk adjusted measure of return on the firm's stock was found to be 0.1398. Test, against a two-sided alternative, the null hypothesis that the population correlation is 0.