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Chapter 6: Problem 60

Stress on the job is a major concern of a large number of people who go into managerial positions. It is estimated that \(80 \%\) of the managers of all companies suffer from job-related stress. a. What is the probability that in a sample of 200 managers of companies, exactly 150 suffer from job-related stress? b. Find the probability that in a sample of 200 managers of companies, at least 170 suffer from job-related stress. c. What is the probability that in a sample of 200 managers of companies, 165 or fewer suffer from job-related stress? d. Find the probability that in a sample of 200 managers of companies, 164 to 172 suffer from job-related stress.

Short Answer

Expert verified
The probabilities for each situation depend on the precise calculations mentioned above. Given the considerable workload, actual numeric solutions are best left to a calculator or statistics software.

Step by step solution

01

Calculation for exactly 150 managers suffering from stress

To calculate this, use the binomial probability formula: \( P(r; n, p) = C(n, r) \cdot p^r \cdot (1 - p)^{n - r} \). Here, n = 200, r = 150, and p = 0.8. \(C(n,r)\) refers to the combination of n items taken r at a time, and can be calculated as \(n!/(r!(n - r)!) \). Using these values in the formula, calculate the desired probability.
02

Calculation for at least 170 managers suffering from stress

This involves calculating the probability of 170, 171, ..., up to 200. You can calculate these probabilities individually and then add them up, but alternatively, you can use the property of binomial distribution that the sum of all probabilities equals to 1. Hence, the probability of 'at least 170' can be calculated as 1 minus the probability of 'up to 169'. So calculate the probabilities for r = 0 up to r = 169 and add them up, then subtract this sum from 1.
03

Calculation for 165 or fewer managers suffering from stress

This problem follows the same trend as part b except the number is now 165. So we are calculating the sum of binomial probabilities for r = 0 up to r = 165.
04

Calculation for 164 to 172 managers suffering from stress

This problem involves calculating the sum of binomial probabilities for r = 164 up to r = 172. Calculate the probabilities for each of these values of r and add them up.

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Most popular questions from this chapter

Johnson Electronics makes calculators. Consumer satisfaction is one of the top priorities of the company's management. The company guarantees the refund of money or a replacement for any calculator that malfunctions within two years from the date of purchase. It is known from past data that despite all efforts, \(5 \%\) of the calculators manufactured by this company malfunction within a 2-year period. The company recently mailed 500 such calculators to its customers. a. Find the probability that exactly 29 of the 500 calculators will be returned for refund or replacement within a 2 -year period. b. What is the probability that 27 or more of the 500 calculators will be returned for refund or replacement within a 2 -year period? c. What is the probability that 15 to 22 of the 500 calculators will be returned for refund or replacement within a 2 -year period?

Determine the following probabilities for the standard normal distribution. a. \(P(-1.83 \leq z \leq 2.57)\) b. \(P(0 \leq z \leq 2.02)\) c. \(P(-1.99 \leq z \leq 0)\) d. \(P(z \geq 1.48)\)

Suppose you are conducting a binomial experiment that has 15 trials and the probability of success of \(.02\). According to the sample size requirements, you cannot use the normal distribution to approximate the binomial distribution in this situation. Use the mean and standard deviation of this binomial distribution and the empirical rule to explain why there is a problem in this situation. (Note: Drawing the graph and marking the values that correspond to the empirical rule is a good way to start.)

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