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Chapter 7: Problem 40

Johnson Electronics Corporation makes electric tubes. It is known that the standard deviation of the lives of these tubes is 150 hours. The company's research department takes a sample of 100 such tubes and finds that the mean life of these tubes is 2250 hours. What is the probability that this sample mean is within 25 hours of the mean life of all tubes produced by this company?

Short Answer

Expert verified
The probability that the sample mean is within 25 hours of the mean life of all tubes produced by this company is 0.905.

Step by step solution

01

Calculate the z-score for +25 hours

First, we will calculate the z-score for +25 hours from the mean. Using the Z-score formula: z = (x - 渭) / (蟽/鈭歯), where x is the sample mean (2250), 渭 is the population mean (which in this case will be the sample mean 卤 25 hours, therefore 2275), 蟽 is the standard deviation (150) and n is the size of the sample (100). By substituting these values into the formula, we find the z-score to be -1.67.
02

Calculate the z-score for -25 hours

Second, we will calculate the z-score for -25 hours from the mean. Utilising the same formula, the only thing changing here is the value for 渭 which, in this case, will be 2225. Substituting these values into the formula, we find the z-score to be 1.67.
03

Calculate the probability

Having the z-score calculated, we can now find out the probability. We need to find the probability that z lies between -1.67 and 1.67. Looking these values up in the z-table, we find that the probabilities are 0.9525 and 0.0475 respectively. Therefore, the probability that this sample mean is within 25 hours of the mean life of all tubes produced by this company is 0.9525 - 0.0475 = 0.905.

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