1. If the random variable z is the standard normal score and P(z < a) > 0.5, then a > 0. Why or why not?
2. Given a binomial distribution with n = 27 and p = 0.81, would the normal distribution provide a reasonable approximation? Why or why not?
3. Find the area under the standard normal curve for the following:
(A) P(z < -1.13)
(B) P(0 < z < 1.29)
(C) P(-0.89 < z < 0.91)
4. Find the value of z such that approximately 27.64% of the distribution lies between it and the mean.
5. Assume that the average annual salary for a worker in the United States is $33,000 and that the annual salaries for Americans are normally distributed with a standard deviation equal to $6,250. Find the following:
(A) What percentage of Americans earn below $21,000?
(B) What percentage of Americans earn above $41,000?
Please show all of your work.
6. X has a normal distribution with a mean of 80.0 and a standard deviation of 4.0. Find the following probabilities:
(A) P(x < 75.0)
(B) P(75.0 < x < 85.0)
(C) P(x > 87.0) (Points : 6)
7. Answer the following:
(A) Find the binomial probability P(x = 4), where n = 14 and p = 0.60.
(B) Set up, without solving, the binomial probability P(x is at most 4) using probability notation.
(C) How would you find the normal approximation to the binomial probability P(x = 4) in part A? Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations.