Sample Variability Consider 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.

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Presentation transcript:

Sample Variability Consider 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, ū 1 = (0+2)/2 = 1 and take another sample and get a sample mean ū 2 = (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 =.08 P(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:

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 1 LOWER BOUND = 38.5 UPPER BOUND = 40 MEAN =39 ANSWER: Section 7.3, Page 147

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 2 AREA FROM LEFT = 0.05 MEAN = 39 ANSWER: NORMDIST 2 AREA FROM LEFT =.95 MEAN = 39 ANSWER: 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 Sampling Distribution

Problems 9Problems, Page 149

Problems 10Problems, Page 150

Problems 11Problems, Page 151