(b) Find the probability that the sample standard deviation is less than $1,500.

(10%)5. Independent random samples were taken of male and female clients of University Entrepreneurship Centers. These clients were considering starting a business. Of 94 male clients, 53 actually started a business venture, as did 47 of 68 female clients. Find and interpret the p-value of a test of equality of the population proportions against the alternative that the proportion of female clients actually starting a business is higher than the proportion of male clients.

(20%)6. Supporters claim that a new windmill can generate an average of at least 800 kilowatts of power per day. Daily power generation for the windmill is assumed to be normally distributed with a standard deviation of 120 kilowatts. A random sample of 100 days is taken to test this claim against the alternative hypothesis that the true mean is less than 800 kilowatts. The claim will be accepted if the sample mean is 776 kilowatts or more and rejected otherwise. (a) What is the probability \(\alpha\) of a Type I error using the decision rule if the population mean is in fact 800 kilowatts per day?

(b) What is the probability \(\beta\) of a Type II error using this decision rule if the population mean is in fact 740 kilowatts per day?

(c) Suppose that the same decision rule is used, but with a sample of 200 days rather than 100 days. (i)Would the value of \(\alpha\) be larger than, smaller than, or the same as that found in (a)?

(ii)Would the value of \(\beta\) be larger than, smaller than, or the same as that found in (b)?

(d) Suppose that a sample of 100 observations was taken, but that the decision rule was changed so that the claim would be accepted if the sample mean was at least 765 kilowatts. (i)Would the value of \(\alpha\) be larger than, smaller than, or the same as that found in (a)?

(ii)Would the value of \(\beta\) be larger than, smaller than, or the same as that found in (b)?

(15%)7. 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 $$b_0 = 10.1 \text{ and } b_1 = 8.4$$ The regression and error sums of squares were $$SSR = 128 \text{ and } SSE = 286$$ (a) Find and interpret the coefficient of determination.

(b) Test at the \(10\%\) significance level against a two-sided alternative the null hypothesis that \(\beta_{1}\) is 0.

(c) Find

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