(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?

III. (25%) Suppose that the starting salaries of female managers have a positively skewed distribution with mean of $56,000 and a standard deviation of $12,000. The starting salaries of male managers are positively skewed with a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female managers and a random sample of 40 male managers are selected. (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.

1. (15%) Let X₁, X₂, X₃ denote a random sample of size n = 3 from a distribution with the p.d.f. $$f(x) = \binom{2}{3} \binom{3}{2}^{x-1}, x=1, 2, 3, \dots.$$ That is, X₁, X₂, X₃ are mutually independent and each has the geometric distribution. (a) (5%) Compute \( \mathrm{P}(\mathrm{X}1 = 1, \mathrm{X}2 = 2, \mathrm{X}_3 = 1) \).

(b) \((5\%)\) Determine \(\mathrm{P}(\mathrm{X}1 + \mathrm{X}2 + \mathrm{X}_3 = 4)\).

(c) (5%) If Y equals the maximum of \( X{1}, X{2}, X{3} \), find \( P(Y \leq 2) = P(X{1} \leq 2)P(X{2} \leq 2)P(X{3} \leq 2) \). (Please show the process of calculation and round to four decimal places)

2. (15%) Let X₁, X₂, ..., Xₙ be a random sample of size n from uniform distribution on interval (θ-1, θ+1). (a) (5%) Find the method of moments estimator for \(\theta\).

(b) \((5\%)\) Is your estimator in part (a) an unbiased estimator for \(\theta\)

(c) (5%) Given the following \( n = 5 \) observations of \( X \), give a point estimate of \( \theta \): 6.6, 7.7, 6.9, 8.3, 7.5. (Please show the process of calculation)

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