Sample VariabilityConsider the small population of integers {0, 2, 4, 6, 8}It is clear that the mean, μ = 4. Suppose we did not know the population mean and wanted to estimate it with a sample mean with sample size 2. (We will use sampling with replacement)

We take one sample and get sample mean, ū1 = (0+2)/2 = 1 and take another sample and get a sample mean ū2 = (4+6)/2 = 5.

Why are these sample means different?

Are they good estimates of the true mean of the population?

What is the probability that we take a random sample and get a sample mean that would exactly equal the true mean of the population?

1Section 7.1, Page 137

Sampling Distribution

Each of these samples has a sample mean, ū. These sample means respectively are as follows:

P(ū = 1) = 2/25 = .08P(ū = 4) = 5/25 = .20

2Section 7.1, Page 138

Sampling Distribution

3Section 7.1, Page 138

Shape is normal

Mean of the sampling distribution = 4, the mean of the population

Sampling Distributions and Central Limit Theorem

4Section 7.2, Page 141

Sample sizes ≥ 30 will assure

a normal distribution.

Alternate notation:

€

SE(x )

Central Limit Theorem

5Section 7.2, Page 144

Central Limit Theorem

6Section 7.2, Page 145

Calculating Probabilities for the Mean

Kindergarten children have heights that are approximately normally distributed about a mean of 39 inches and a standard deviation of 2 inches. A random sample of 25 is taken. What is the probability that the sample mean is between 38.5 and 40 inches?

P(38.5 < sample mean <40) =NORMDIST 1LOWER BOUND = 38.5UPPER BOUND = 40MEAN =39

ANSWER: 0.8881

7Section 7.3, Page 147

€

=2 / 25 = 0.4

€

SE(x )

Calculating Middle 90%

Kindergarten children have heights that are approximately normally distributed about a mean of 39 inches and a standard deviation of 2 inches. A random sample of 25 is taken. Find the interval that includes the middle 90% of all sample means for the sample of kindergarteners.

NORMDIST 2AREA FROM LEFT = 0.05MEAN = 39 ANSWER: 38.3421

NORMDIST 2AREA FROM LEFT = .95MEAN = 39 ANSWER: 39.6579

The interval (38.3 inches, 39.7 inches) contains the middle 90% of all sample means. If we choose a random sample, there is a 90% probability that it will be in the interval.

8Section 7.3, Page 147

€

2 / 25 = 0.4

€

2 / 25 = 0.4

Sampling Distribution

€

ux = 39

€

σ x =2

25

€

SE(x ) =

€

SE(x ) =

Problems

9Problems, Page 149

Problems

10Problems, Page 150

Problems

11Problems, Page 151