Binomial Probability

1. If the random variable z is the standard normal score, is it true that P(z < 3) could easily be approximated without referring to a table? Why or why not? (Points : 3) 

2. Given a binomial distribution with n = 36 and p = 0.87, would the normal distribution provide a reasonable approximation? Why or why not? (Points : 3) 

3. Find the area under the standard normal curve for the following: (A) P(z < -0.25) (B) P(0 < z < 0.55 ) (C) P(-1.91 < z < 1.06) (Points : 6) 

4. Find the value of z such that approximately 30.78% of the distribution lies between it and the mean. (Points : 3) 

5. Assume that the average annual salary for a worker in the United States is $41,000 and that the annual salaries for Americans are normally distributed with a standard deviation equal to $7,000. Find the following: (A) What percentage of Americans earn below $27,000? (B) What percentage of Americans earn above $43,000? Please show all of your work. (Points : 6) 

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 < 79.0) (B) P(78.0 < x 86.0) (Points : 6) 

7. Answer the following:

(A) Find the binomial probability P(x = 4), where n = 12 and p = 0.70.

(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. (Points : 6)