The answer to Normal distribution problems
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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.
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Solution Summary
Normal distribution questions are solved. Binomial probability is used for the notation and for the normal approximation.
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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?
Answer:
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)
Answers:
(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 ...
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