3. Suppose that each of two investments for a portfolio has a \( 4\% \) chance of a loss of $5 million, a \( 2\% \) chance of a loss of $1 million, and \( 94\% \) chance of a gain of $2 million. Two investments are independent of each other. The 10th percentile for the distribution of this portfolio investment return is __. (5%)

4. Consider a random sample \( \{x_1, \dots, x_n\} \) from the distribution with mean \( \mu \) and known variance \( \sigma^2 \). Define \( \bar{x} = n^{-1} \sum_{i=1}^{i=n} x_i \), \( s_1^2 = (n-2)^{-1} \sum_{i=1}^{i=n} (x_i - \mu)^2 \), and \( s_2^2 = (n-2)^{-1} \sum_{i=1}^{i=n} (x_i - \bar{x})^2 \). If n=60, the distributions of \( (n-2)s_1^2/\sigma^2 \) and \( (n-2)s_2^2/\sigma^2 \) are ________ and ________, respectively. (10%) (Justify your answers)

5. Consider a population of five weights identical in appearance but weighing 1, 3, 5, 7 and 9 ounces. The variance for the population (\( \sigma^2 \)) is ________. (5%) Sampling without replacement from the above population with a sample size of 2 produces ten possible samples. Using the ten sample mean values to calculate the variance of \( \bar{X} \), it is ________. (5%) Why is this value not the same as \( \sigma^2/n \)? ________ (5%)

(1) Conduct a statistical test to determine whether there is a significant difference between the variances in the SAT score of male and female students. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(2) Based on the result of previous question, conduct a statistical test to determine whether the mean SAT score of male students is significantly LOWER than the mean SAT score of female students. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(3) Are the proportions of in-state students the same for male and female students? Report the test statistic and P-value. (5%)

(4) Construct a \( 95\% \) confidence interval for the difference of the proportion of in-state students for male and female students. \( (5\%) \)

(1) Find the within-group variance. (5%)

(2) To test the claim that there is no difference among the means in blood pressure reduction of three groups, what is the test statistics? (5%)

(1) Conduct a test to determine whether there is at least one significant predictor. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(2) Conduct a test to determine whether smoking a significant predictor for the risk of a stroke. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

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