2. [15%] A random sample of 60 observations from a normally distributed population possesses a mean equal to 37.2 and a standard deviation equal to 4.3.

(a) Please find a \(95\%\) confidence interval for \(\mu\).

(b) What do you mean when you say that a confidence coefficient is 0.95?

(c) What happens to the width of a confidence interval as the value of the confidence coefficient is decreased while the sample size is held fixed?

3. [10%] A statistician reported to a car insurance company a confidence interval for the proportion (p) of convertible cars that had been involved in major accidents during the past year. The 95% confidence interval for p was reported to be the interval from 0.16 to 0.44. (a) What is the statistician's estimate of \( p \)?

(b) Approximately what sample size did the statistician use?

4. [30%] Based on the following 10 pairs of data: | X | 6 | -3 | 4 | 3 | -2 | -4 | 0 | -1 | -5 | 2 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Y | 10 | 1 | 8 | 7 | 1 | 2 | 5 | 3 | 2 | 6 | and assuming the following linear regression model, Y = α + βX + ε, where ε has zero mean and variance equal to σ². Please answer the following questions: (a) Compute the least-squares point estimator of \(\alpha\) and \(\beta\).

(b) Compute the coefficient of determination.

(c) Calculate the variance estimates for \(\hat{\beta}\).

(d) With \(10\%\) significance level, test the hypothesis " \(H_0: \sigma^2 = 1\) . Explain your answer in details.

5. [15%] Suppose an employer randomly selects three new employees from a total of ten applicants, six men and four women. Let X be the number of women who are hired. (a) Find the mean and variance of \( X \).

(b) Find the probability that no women are hired.

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