/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 The GPAs of all students enrolle... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 49

The GPAs of all students enrolled at a large university have an approximately normal distribution with a mean of \(3.02\) and a standard deviation of \(.29 .\) Find the probability that the mean GPA of a random sample of 20 students selected from this university is a. \(3.10\) or higher b. \(2.90\) or lower c. \(2.95\) to \(3.11\)

Short Answer

Expert verified
The probability that the mean GPA of a random sample of 20 students selected is \na) 3.10 or higher is 0.1075, \nb) 2.90 or lower is 0.0314, and \nc) between 2.095 and 3.11 is 0.7776.

Step by step solution

01

- Understanding given data

The mean of the GPAs (\(渭\)) is given as 3.02 and their standard deviation (\(蟽\)) is said to be 0.29. A random sample of 20 students is selected from this university.
02

- Calculating Standard Error

We first calculate standard error (\(SE\)), which is given by \(蟽/鈭歯\), where \(n\) is the sample size. Since we know that \(蟽 = 0.29\) and \(n = 20\), we can calculate \(SE\) to be \(0.29/鈭�20 = 0.0648\).This value will be needed to calculate Z scores.
03

- Calculating Z scores

The Z-score is calculated by \((x - 渭)/SE\), where \(x\) is the value for which the score is calculated. \nTo find the probability that the mean GPA is 3.10 or higher, we find its Z score using \((3.10 - 渭)/SE = (3.10 - 3.02) / 0.0648 = 1.24\). \nFor a GPA of 2.90 or lower, the Z score will be \((2.90 - 渭)/SE = (2.90 - 3.02) / 0.0648 = -1.86\). \nFor the range of 2.95 to 3.11, our relevant Z scores will be \((2.95 - 渭)/SE = (2.95 - 3.02) /0.0648 = -1.08\) and \((3.11 - 渭)/SE = (3.11 - 3.02) /0.0648 = 1.39\).
04

- Finding Probabilities

We can now use a standard normal distribution table to find the associated probabilities. For a Z-score of 1.24, the probability from the table is 0.8925. Therefore, the probability of having a GPA of 3.10 or higher is \(1 - 0.8925 = 0.1075\). \nFor the Z score -1.86, the probability is 0.0314, so probability of obtaining 2.90 GPA or less is simply 0.0314. \nFinally, for the range \(2.95 鈮� GPA 鈮� 3.11\), we find probabilities for the Z scores -1.08 and 1.39. The associated probabilities are 0.1401 and 0.9177 respectively. The probability that GPA is between 2.95 and 3.11 is \(0.9177 - 0.1401 = 0.7776\).

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