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Distributions of Sample Means and Sample Proportions

Distributions of Sample Means and Sample Proportions. BUSA 2100, Sections 7.0, 7.1, 7.4, 7.5, 7.6. Point Estimates. We don’t expect the sample mean to be exactly equal to the population mean.

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Distributions of Sample Means and Sample Proportions

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  1. Distributions of Sample Means and Sample Proportions BUSA 2100, Sections 7.0, 7.1, 7.4, 7.5, 7.6

  2. Point Estimates • We don’t expect the sample mean to be exactly equal to the population mean. • Even when two samples are selected from the same population, the two sample means will be different! • In fact, there is an entire distribution of different sample means from the same population. • The distribution of all sample means for samples of a fixed size is called the distribution of sample means.

  3. Distribution of Sample Means • Ex.: A distribution of 25 individual items (population) is: { 40, 50, 55, 59, 62, 64, 65, 66, 67, 68, 69, 70, 70, 70, 71, 72, 73, 74, 75, 76, 78, 81, 85, 90, 100 } . • How many samples of size 4 are possible? • We will take only 12 samples of size 4, and calculate the mean of each sample.

  4. Distribution of Sample Means Example, Page 2 .

  5. Distribution of Sample Means (for all samples of a fixed size) • Suppose all samples of size 4 had been chosen.

  6. Central Limit Theorem • Central Limit Theorem: For large values of n, the distribution of sample means becomes normally distributed, regardless of the shape of the distribution of individual items (population). (see text) • When n is larger than 30, that is considered large enough.

  7. Distribution of Sample Means (Summary) • The distribution of sample means for all samples of a fixed size n (from a population with mean mu and standard deviation sigma) has mean mu and standard deviation sigma / sqrt (n). • The symbol for the std. deviation of the sample means is sigma sub X-bar. • Relate to the size of n.

  8. z-Formulas .

  9. Salaries Example • Example: For a population of 2,000 management executives, the salaries are normally distributed with a mean of $56,000 and a standard deviation of $4,200. • A sample of 36 managers is selected and the mean salary is calculated. • What is the probability that the sample mean is within $500 of the population mean? • In other words, what is the probability that the sampling error is <= $500?

  10. Salaries Example, Page 2 .

  11. Salaries Example, Page 3 .

  12. Distrib. of Sample Proportions • Proportions are always between 0 & 1. • Proportions are binomial. • A sample proportion, p-bar, is a point estimate for the population proportion, p . • For a population (distribution of individual items) with proportion p, the distribution of sample proportions for all samples of a fixed size n has mean = p, and std. dev. = sigma sub p-bar =sqrt [ p * (1 - p) / n ]

  13. Customer Proportion Example • Example: Last year, 30 percent of a company’s mail orders came from first-time customers. • A random sample of 80 mail-order customers is selected and the proportion of first-time customers is calculated.

  14. Proportion Example, Page 2 • What is the probability that the sample proportion is within 4% (.04) of the population proportion?

  15. Proportion Example, Page 3 • Part (b): Same question for n = 250.

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