(a). In the united states, about 20% of adults have four -year college degree (The wall street Journal, April 26,2012). Suppose four adults are randomly selected

1. what is the probability that all four adults have a college degree?

2. What is the probability that at most two of the four adults have a college degree?

3. Let X denote the number of four-year college degree holders among the four randomly selected adults. Calculate the expected value, variance, and standard deviation of X: that is, find E(X), Var(X), and SD (X).

(b) Scores on a management aptitude examination are normally distributed with a mean of 72 and a standard deviation of 8.

1. what is the probability that a randomly selected manager will score above 70?

2. what is the lowest score that will place a manager in the top 2.5% & of the distribution?

(c) A random sample of 36 observations has been drawn from a normal distribution with a mean of 50 and a standard deviation of 12.

1. find the probability distribution of the sample mean x

2. find the probability that the sample mean x is in the interval 47 less than/ equal x less than/ equal 53.

3. Now, if we have a sample of 36 observations with a mean of 48, the population mean is unknown, while the population standard deviation is still 12.

a). What would be a good estimate of ? Why?

b). calculate the 95% confidence interval for

c). What is the probability that falls into the interval in part (b) above, and why?

d). Now, suppose you add 4 identical numbers of 48 into the original sample of 36 observations with a mean of 48; nothing else changes. Calculate the 95% confidence interval for .