(b) \((5\%)\) Find \(E\left(3X^{2} - 2X + 4\right)\).

1.2 (10%) It is claimed that 15% of the ducks in a particular region have patent schistosome infection. Suppose that seven ducks are selected at random. Let X equal the number of ducks that are infected. (a) \((5\%)\) Find \(\mathrm{P}(\mathrm{X} = 1)\)

(b) \((5\%)\) Find \(\mathrm{P}(\mathrm{X}\geq 2)\)

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.

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