Assignment:
Q1. 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?
Q2. Given a binomial distribution with n = 20 and p = 0.26, would the normal distribution provide a reasonable approximation? Why or why not?
Q3. 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)
Q4. 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?
Q5. Find the value of z such that approximately 47.93% of the distribution lies between it and the mean.
Q6. 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)
Q7. 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.
Provide complete and step by step solution for the question and show calculations and use formulas.