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