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Chapter 8

Sampling Distributions of a Sample Mean

Section 2

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When the objective of a statistical investigation is to make an inference about the population mean µ, it is natural to consider the sample mean as an estimate of µ.

To understand how inferential procedures based on the sample mean work, we must first study how sampling variability causes the sample mean to vary in value from one sample to the next. The behavior of the sample mean is described by its sampling distribution.

BIG Idea

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Sampling Distribution

The distribution of a statistic is called its sampling distribution.

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Example

Consider a population that consists of the numbers 1, 2, 3, 4 and 5 generated in a manner that the probability of each of those values is 0.2 no matter what the previous selections were. This population could be described as the outcome associated with a spinner such as given below. The distribution is next to it.

x p(x)1 0.22 0.23 0.24 0.25 0.2

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Example

If the sampling distribution for the means of samples of size two is analyzed, it looks like

Sample Sample

1, 1 1 3, 4 3.51, 2 1.5 3, 5 41, 3 2 4, 1 2.51, 4 2.5 4, 2 31, 5 3 4, 3 3.52, 1 1.5 4, 4 42, 2 2 4, 5 4.52, 3 2.5 5, 1 32, 4 3 5, 2 3.52, 5 3.5 5, 3 43, 1 2 5, 4 4.53, 2 2.5 5, 5 53, 3 3

frequency p(x)

1 1 0.041.5 2 0.082 3 0.12

2.5 4 0.163 5 0.20

3.5 4 0.164 3 0.12

4.5 2 0.085 1 0.04

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Example

The original distribution and the sampling distribution of means of samples with n=2 are given below.

54321

Original distribution

54321

Sampling distribution

n = 2

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ExampleSampling distributions for n=3 and n=4 were calculated and are illustrated below.

Sampling distribution n = 354321

Sampling distribution n = 454321

8 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Simulations

Means (n=120)432

Means (n=60)432

Means (n=30)432

To illustrate the general behavior of samples of fixed size n, 10000 samples each of size 30, 60 and 120 were generated from this uniform distribution and the means calculated. Probability histograms were created for each of these (simulated) sampling distributions.

Notice all three of these look to be essentially normally distributed. Further, note that the variability decreases as the sample size increases.

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Simulations

Skewed distribution

To further illustrate the general behavior of samples of fixed size n, 10000 samples each of size 4, 16 and 32 were generated from the positively skewed distribution pictured below. It represents the length of overtime games played in the NHL from 1970 – 1993.

Notice that these sampling distributions all all skewed, but as n increased the sampling distributions became more symmetric and eventually appeared to be almost normally distributed.

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Terminology

Let denote the mean of the observations in a random sample of size n from a population having mean µ and standard deviation . Denote the mean value of the distribution by and the standard deviation of the distribution by (called the standard error of the mean), then the rules on the next two slides hold.

x

xx

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Properties of the Sampling Distribution of the Sample Mean.

Rule 2:

This rule is approximately correct as long as no more than 5% of the population is included in the sample.

xn

x

n

x Rule 1:

Rule 3: When the population distribution is normal, the sampling distribution of is also normal for any sample size n.

x

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Central Limit Theorem.

Rule 4: When n is sufficiently large, the sampling distribution of is approximately normally distributed, even when the population distribution is not itself normal.

x

13 © 2008 Brooks/Cole, a division of Thomson Learning, Inc.

Illustrations of Sampling Distributions

Symmetric normal like population

Population

n =4n = 9

n = 16

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Illustrations of Sampling Distributions

Skewed population

Populationn=4n=10n=30

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More about the Central Limit Theorem.

The Central Limit Theorem can safely be applied when n exceeds 30.

If n is large or the population distribution is normal, the standardized variable

has (approximately) a standard normal (z) distribution.

X

X

x xz

n

ExampleHot Dogs!

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A hot dog manufacturer asserts that one of its brands hasan average fat content of µ=18 g per hot dog. Consumers wouldnot be unhappy if the content was less than 18 but would be ifit exceeded 18 grams. Suppose the standard deviation of the xdistribution is 1.

A testing organization is asked to analyze 36 hot dogs.

What is the mean of the sample?

What is the standard deviation of the sample mean?

Example

Suppose that we randomly select a sample of 64 measurements from a population having a mean equal to 20 and a standard deviation equal to 4.

a.) Describe the shape of the sampling distribution of the sample mean

b.) Find the mean and the standard deviation of the sampling distribution of the sample mean.

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x

Hospital Example

The average length of a hospital stay in the US is µ=9 days with a standard deviation of 3 days. Assume a random sample of 100 patients is obtained and the mean stay for the 100 patients is obtained. What is the probability that the average length of stay for this group of patients will be less than 9.6 days?

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Hospital Example Continued

Step 1: Find the mean and standard deviation for the sample.

Step 2: Find the z score for the value of 9.6.

Step 3: Use the standard normal distribution table to find the answer.

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ACT Example

The scores of students on the ACT college entrance exam in a recent year had a Normal distribution with µ=18.6 and a standard deviation of 5.9.

a.) What is the probability that a single student randomly chosen from all those taking the test scores 21 or higher?

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ACT Example Continued

b.) Now take a simple random sample of 50 students who took the test. What is the probability that the mean score of these students is 21 or higher?

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Example

A food company sells “18 ounce” boxes of cereal. Let x denote the actual amount of cereal in a box of cereal. Suppose that x is normally distributed with µ = 18.03 ounces and = 0.05.

a) What proportion of the boxes will contain less than 18 ounces?

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Example - continued

b) A case consists of 24 boxes of cereal. What is the probability that the mean amount of cereal (per box in a case) is less than 18 ounces?

The central limit theorem states that the distribution of is normally distributed sox