1.3 (10%) If X is normally distributed with a mean of 77 and a variance of 25. Let $$\overline{X}$$ be the mean of a random sample of size 16 from this distribution.

(a) \((5\%)\) Find \(P(77 < X < 79.5)\).

(b) \((5\%)\) Find \(P\left(77 < \overline{X} < 79.5\right)\).

1.4 (10%) Suppose the population is normally distributed with unknown standard deviation $$\sigma$$. Consider the following random sample of 4 observations: 25, 47, 32, 56. (a) \((5\%)\) Find a point estimate of \(\sigma\)

(b) \((5\%)\) Find a \(95\%\) confidence interval for \(\mu\).

1.5 (10%) Let X and Y denote the tarsus lengths of male and female grackles, and $$\mu_X$$ and $$\mu_Y$$ are the male and female mean tarsus lengths, respectively. Two samples of size 25 are taken from each populations assumed to be normally with equal variances. The male sample has a mean of 35.5 and standard deviation 3.0 while the female sample has a mean of 33.0 and standard deviation 4.0. We want to test the hypothesis that the mean tarsus lengths are equal against the hypothesis that male has longer tarsus lengths. (a) \((2\%)\) Write the null and alternative hypotheses with symbols.

(b) \((6\%)\) Compute the test statistics.

(c) \((2\%)\) Use 0.05 as significance level and draw a conclusion.

2.2 (10%) Among 100 managers with degrees in business administration, the following data have been accumulated as to their fields of concentration. | Major | Position | | TOTAL | | --- | --- | --- | --- | | | Top Management | Middle Management | | | Management | 28 | 22 | 50 | | Marketing | 12 | 8 | 20 | | Accounting | 10 | 20 | 30 | | TOTAL | 50 | 50 | 100 | To test if the position in management is independent of field (major) of concentration, (a) (5%) State the null and alternative hypotheses.

(b) (5%) At 95% confidence using the p-value approach, test the hypotheses. What is your conclusion? (For the test, give the null and alternative hypotheses, test statistic, and conclusion.)

2.3 (10%)Employees of AAA Corporation are about to undergo a retraining program. Management is trying to determine which of three programs is the best. They believe that the effectiveness of the programs may be influenced by sex. A factorial experiment was designed. You are given the following information of score. Factor B (Sex) | Factor A (Program) | 1 | 2 | | --- | --- | --- | | A | 26 19 20 21 | 24 21 20 23 | | B | 30 24 25 29 | 33 27 31 29 | | C | 26 22 27 17 | 31 23 24 26 | Given SSA=184.33, SSB=28.17, SSAB=8.33, SST=405.83 (a) (5%) Set up the ANOVA table.

(b) (5%) What advice would you give AAA Corporation? Use

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