The answer to Normal distribution problems
1. If the random variable z is the standard normal score and a > 0, is it true that P(z > -a) = P(z < a)? Why or why not?

2. Given a binomial distribution with n = 20 and p = 0.26, 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 < -0.74)
(B) P(-0.87 < z < 0)
(C) P(-2.03 < z < 1.66)

4. Assume that the average annual salary for a worker in the United States is $32,500 and that the annual salaries for Americans are normally distributed with a standard deviation equal to $6,250. Find the following and show all of your work:
(A) What percentage of Americans earn below $21,000?
(B) What percentage of Americans earn above $39,000?

5. Find the value of z such that approximately 47.93% of the distribution lies between it and the mean.

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 > 83.0)

7. Answer the following:
(A) Find the binomial probability P(x = 5), where n = 14 and p = 0.70.
(B) Set up, without solving, the binomial probability P(x is at most 5) using probability notation.
(C) How would you find the normal approximation to the binomial probability P(x = 5) in part A? Please show how you would calculate µ and ? in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations.