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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)

(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 ยต 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.

Solution Preview

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?

Answer: The given statement is true. That is, P (z > -a) = P (z < a) for all a > 0
Explanation:
The standard normal distribution is symmetric about the mean and P (z < 0) = P (z > 0) = 0.5
Also P (z > a) + P (z < a) = 1 for all a
That is, P (z > a) = 1 - P (z < a)
Hence P (z > -a) = P (z < a) for all a > 0

2. Given a binomial distribution with n = 20 and p = 0.26, would the normal distribution provide a reasonable approximation? Why or why not?

Here n = 20, p = 0.26
The conditions that must be satisfied for a normal approximation to the binomial are:
np and n(1 - p) should be greater than 5.
np = 20 * 0.26 = 5.2 > 5
n(1 - p) = 20(1 - 0.26) = 14.8 > 5
Hence the normal distribution would provide a reasonable approximation.

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)

(A) P (z < -0.74)
P (z < -0.74) = 0.2296

(B) P (-0.87 < z < 0)
P (-0.87 < z < 0) = 0.3078

(C) P (-2.03 < z < 1.66)
P (-2.03 < z < 1.66) = 0.9304

4. Assume that the average ...

Solution Summary

Normal distribution questions are solved. Binomial probability is used for the notation and for the normal approximation.

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