8. (10%) Evaluate \( \int x^{2} \sin(1 - x) dx \).

9. (15%) Sketch the region of integration and write an equivalent integral with the order of integration reversed. Then evaluate both integrals. $$\int_0^{3/2} \int_{-\sqrt{9-4y^2}}^{\sqrt{9-4y^2}} y \, dx \, dy.$$

I. (10%) A stock portfolio is composed of two stocks: \( A_{1} \) and \( B_{1} \). Another stock portfolio also has two investments: stocks \( A_{2} \) and \( B_{2} \). Suppose the mean prices \( (\mu) \) and the standard deviations \( (\sigma) \) of \( A_{1} \) and \( A_{2} \) are equal, and the mean price and standard deviation of \( B_{1} \) are equal to those of \( B_{2} \) (i.e., \( \mu_{A_1} = \mu_{A_2}, \mu_{B_1} = \mu_{B_2}, \sigma_{A_1} = \sigma_{A_2}, \sigma_{B_1} = \sigma_{B_2} \)). Suppose each stock comprises 50% of the portfolio. However, the correlation between stocks is higher in the first portfolio (i.e., \( \rho_1 > \rho_2 \)). Explain the difference in expected returns and risks between the two portfolios.

(a) Describe the probability distribution of \( P_A \), the proportion of respondents who was identified with Party A in the sample. Explain whether normal distribution is appropriate to approximate the distribution of \( P_A \).

(b) Suppose \( 20\% \) the population is leaning toward Party A. If the pollster is to interview 10 people, what is the probability that the pollster does not find anyone who is leaning toward Party A?

(c) Suppose the pollster interviewed 765 people and found that 407 of the respondents were affiliated with Party A. Can the pollster conclude at \( 5\% \) significance level that Party A has more than \( 50\% \) of support from the public? Explain your answer.

(d) Suppose the pollster wants to estimate \( p_A \) within 0.03 margin (i.e. \( \pm 3\% \)) with \( 95\% \) confidence. How many people should the pollster interview?

(a) Explain what can we infer from the description that the salaries are "positively skewed"?

(b) Describe the sampling distribution of the differences between the average salaries of male and female managers, including its expected value and standard error.

(c) What is the probability that the sample mean salary of male managers will not exceed that of the female managers?

IV. (15%) The city has recently installed red light cameras to discourage traffic signal violations. The Department of Transportation has collected the number of violations for the week prior to the installation of the camera (the first row) and the week after the installation (the second row). Based on the data, can we conclude at 5% significance level that the camera can reduce traffic light violations? Explain your answer.

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