Chapter 7, Problem 1P

### Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570

Chapter
Section

### Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570
Textbook Problem

# Describe the distribution of sample means (shape, expected value, and standard error) for samples of n = 100 selected from a population with a mean of μ = 40 and a standard deviation of σ = 10.

To determine

To define: The distribution of sample means.

The distribution of the sample mean is approximately normal with mean of μ=40 and the standard error of σM=1.

Explanation

Given Info:

The population mean is μ=40, the standard deviation is σ=10 and the sample size is n=100.

Calculation:

Since, given sample size n=100 is greater than 30, so according to central limit theorem, the distribution of sample means is approximately normal. Therefore, the shape of the distribution of sample means is normal.

Since expected value of the mean of the distribution of sample means is the average of all the possible samples means. The sample means are the representative of the population mean from which samples have been drawn. The average of all possible samples takes all population units into the consideration. Therefore, average of all sample means will be equal to population mean. Hence, sample means is equals to the population mean that is μ=40.

If μ and σ represents the population mean and standard deviation respectively. Let n represents sample size. Then,

μ=40σ=10n=100

Standard error measures the distance expected on an average between M and μ. It is represented by σM and is calculated as:

σM=σn

Substitute σ=10 and n=100

σM=10100=1010=1

So, standard error of distribution of sample means is σM=1.

Thus, the shape, mean and standard error of the distribution of sample means are normal, 40 and 1 respectively.

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