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Chapter 7: Problem 44

Let \(x\) be a continuous random variable that has a normal distribution with \(\mu=75\) and \(\sigma=14\). Assuming \(n / N \leq .05\), find the probability that the sample mean, \(\bar{x}\), for a random sample of 20 taken from this population will be a. between \(68.5\) and \(77.3\) b. less than \(72.4\)

Short Answer

Expert verified
Following these steps allows the chance of the sample mean being within the indicated ranges to be found. Calculations are required in each part of the problem for the accurate values. The answer to part a is \(P_{Z_{77.3}} - P_{Z_{68.5}}\) and the answer to part b is \(P_{Z_{72.4}}\).

Step by step solution

01

Calculate z-scores for parts (a) and (b)

The formula for calculating the z-score is: Z = (\(\overline{X}\) - \(\mu\)) / ( \(\sigma / \sqrt{n}\)). Here, \(\overline{X}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size. (a)...The z-scores for \(68.5\) and \(77.3\) are:\(Z_{68.5}\) = (68.5 - 75) / (14 / \(\sqrt{20}\)) \(Z_{77.3}\) = (77.3 - 75) / (14 / \(\sqrt{20}\))(b)...The z-score for \(72.4\) is:\(Z_{72.4}\) = (72.4 - 75) / (14 / \(\sqrt{20}\))
02

Consult the standard normal distribution table

The standard normal distribution table (often known as the z-table) gives the probability of a Z value being less than or equal to a given value. (a)...To find the probability that \(\overline{x}\) is between \(68.5\) and \(77.3\), the table is consulted based on computed \(Z_{68.5}\) and \(Z_{77.3}\) values. The desired probability P(\(68.5 < \overline{x} < 77.3\)) is the difference in the probabilities: \(P_{Z_{77.3}} - P_{Z_{68.5}}\).(b)...To find the probability that \(\overline{x}\) is less than \(72.4\), the table is consulted based on the calculated \(Z_{72.4}\) value. The probability \(P_{Z_{72.4}}\) gives the answer.
03

Interpret the results

(a) The probability of finding a sample of 20 where the mean score is between 68.5 and 77.3 is \(P_{Z_{77.3}} - P_{Z_{68.5}}\).(b) The probability of finding a sample of 20 where the mean score is less than 72.4 is \(P_{Z_{72.4}}\).

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