b. If the two-sample t test is incorrectly used to analyze paired data, what will happen to the t value if $\rho > 0$ (underestimated or overestimated)? Justify your answer.

II. (50%) 1. (15%) Suppose $X$, $Y$, and $Z$ are independent Poisson random variables, each with mean 1. Find a. $P(X+Y+Z=4)$ b. $E[(X+Y)^2]$ c. $P\left(X=1 \mid \frac{X+Y}{2} = 2\right)$

2. (10%) Suppose that a random sample is to be taken from a Bernoulli distribution for which the value of the parameter p is unknown. Suppose also that it is believed that the value of p is in the neighborhood of 0.2. a. How large a random sample must be taken in order that $P(|\bar{X}_n - p| \leq 0.1) \geq 0.75$ when $p=0.2$?

b. How large a random sample must be taken in order that $E(|\bar{X}_n - p|^2 \leq 0.01)$ when $p=0.2$?

3. (15%) Suppose that a random sample $X_1, \ldots, X_n$ is drawn from a normal distribution for which the value of the mean $\mu$ is unknown and the standard deviation is 2, and the following simple hypotheses are to be tested: $$H_0: \mu = -1$$ $$H_1: \mu = 1$$ Determine the minimum value of $\alpha + \beta$ (type I error+ type II error) that can be attained for each of following values of the sample size n: a. n=1 b. n=16

4. (10%) A study classified each of 134 lawyers into one of four groups, based on observation and an interview. The 62 lawyers in group A were categorized as having high levels of stimulation and support and average levels of public spirit. The 52 lawyers in group B had low stimulation, average support, and high public spirit. Group C contained 7 lawyers, with average stimulation, low support, and low public spirit. The 13 lawyers in group D were assessed as low on all three criteria. Salary levels for these four groups were compared. The sample means were 7.87 for group A, 7.47 for group B, 5.14 for group C, and 3.69 for group D. The F ratio calculated from these data was 25.60. a. Set out complete analysis of variance table.

b. Test the null hypothesis that the population mean salaries are the same for lawyers in these four groups at the 1% level.

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