Introduction to the Practice of Statistics: w/CrunchIt/EESEE Access Card
Introduction to the Practice of Statistics: w/CrunchIt/EESEE Access Card
8th Edition
ISBN: 9781464158933
Author: David S. Moore, George P. McCabe, Bruce A. Craig
Publisher: W. H. Freeman
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Chapter 5, Problem 76E
To determine

To find: The probability that sample mean x¯ is greater than 70.

Expert Solution & Answer
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Answer to Problem 76E

Solution: The probability is 0.02103.

Explanation of Solution

Calculation: The sampling distribution of sample mean x¯ is distributed normallywith mean μx¯=μ and standard deviation σx¯=σ/n, where a random sample is drawn from a population with μ and σ, and the sample size is relatively large.

The value of the mean and the standard deviation of the sampling distribution is calculated as:

μx¯=μ=68.0

And value of σx¯ for the sampling distribution of x¯ can be calculated as:

σx¯=σn=22500=0.9839

The sampling distribution of x¯ is approximately normal with mean μx¯=68.0 and standard deviation σx¯=0.9839.

The probability that x¯ is greater than 70 is the area to the right of the point 70 under the normal curve. To find the area under the normal curve, first, convert the value of sample mean x¯=70 to a Z-score, and then obtain the area by subtracting the area left to the Z-score from 1. The Z-score for the value 70is calculated as:

Z=x¯μx¯σx¯=7068.00.9839=2.033

The area left to the particular Z-score can be obtained using Excel. The following procedure is carried out in Excel:

Step 1: Open the Excel spreadsheet.

Step 2: Type =NORM.DIST() in one of the cells and specify x=2.033,Mean = 0, standard deviation = 1, and cumulative as TRUE. The screenshot is shown below:

Introduction to the Practice of Statistics: w/CrunchIt/EESEE Access Card, Chapter 5, Problem 76E

The area under the standard normal curve to the left of Z=2.033 is obtained as 0.978974from Excel. Therefore, the required probability is calculated as:

10.978974=0.02103

Hence, the probability that x¯ is greater than 70 is 0.02103.

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Chapter 5 Solutions

Introduction to the Practice of Statistics: w/CrunchIt/EESEE Access Card

Ch. 5.1 - Prob. 11ECh. 5.1 - Prob. 12ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - Prob. 18ECh. 5.1 - Prob. 19ECh. 5.1 - Prob. 20ECh. 5.1 - Prob. 21ECh. 5.1 - Prob. 22ECh. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Prob. 27ECh. 5.1 - Prob. 28ECh. 5.1 - Prob. 29ECh. 5.1 - Prob. 30ECh. 5.1 - Prob. 31ECh. 5.1 - Prob. 32ECh. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.2 - Prob. 35UYKCh. 5.2 - Prob. 36UYKCh. 5.2 - Prob. 37UYKCh. 5.2 - Prob. 38UYKCh. 5.2 - Prob. 39UYKCh. 5.2 - Prob. 40UYKCh. 5.2 - Prob. 41UYKCh. 5.2 - Prob. 42UYKCh. 5.2 - Prob. 43UYKCh. 5.2 - Prob. 44UYKCh. 5.2 - Prob. 45UYKCh. 5.2 - Prob. 46ECh. 5.2 - Prob. 47ECh. 5.2 - Prob. 48ECh. 5.2 - Prob. 49ECh. 5.2 - Prob. 50ECh. 5.2 - Prob. 51ECh. 5.2 - Prob. 52ECh. 5.2 - Prob. 53ECh. 5.2 - Prob. 54ECh. 5.2 - Prob. 55ECh. 5.2 - Prob. 56ECh. 5.2 - Prob. 57ECh. 5.2 - Prob. 58ECh. 5.2 - Prob. 59ECh. 5.2 - Prob. 60ECh. 5.2 - Prob. 61ECh. 5.2 - Prob. 62ECh. 5.2 - Prob. 63ECh. 5.2 - Prob. 64ECh. 5.2 - Prob. 65ECh. 5.2 - Prob. 66ECh. 5.2 - Prob. 67ECh. 5.2 - Prob. 68ECh. 5.2 - Prob. 69ECh. 5.2 - Prob. 70ECh. 5.2 - Prob. 71ECh. 5.2 - Prob. 72ECh. 5.2 - Prob. 73ECh. 5.2 - Prob. 74ECh. 5.2 - Prob. 75ECh. 5 - Prob. 76ECh. 5 - Prob. 77ECh. 5 - Prob. 78ECh. 5 - Prob. 79ECh. 5 - Prob. 80ECh. 5 - Prob. 81ECh. 5 - Prob. 82ECh. 5 - Prob. 83ECh. 5 - Prob. 84ECh. 5 - Prob. 85ECh. 5 - Prob. 86ECh. 5 - Prob. 87ECh. 5 - Prob. 88ECh. 5 - Prob. 89ECh. 5 - Prob. 90E
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