Chapter 7: Introduction to Sampling Distributions

Chapter 7 Introduction to Sampling Distributions

Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania

Terms, Statistics & Parameters • Terms: Population, Sample, Parameter, Statistics

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Why Sample? • At times, we’d like to know something about the population, but because our time, resources, and efforts are limited, we can take a sample to learn about the population.

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Types of Inference 1) Estimation: We estimate the value of a population parameter. 2) Testing: We formulate a decision about a population parameter. 3) Regression: We make predictions about the value of a statistical variable.

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Sampling Distributions • To evaluate the reliability of our inference, we need to know about the probability distribution of the statistic we are using. • Typically, we are interested in the sampling distributions for sample means and sample proportions.

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The Central Limit Theorem (Normal) • If x is a random variable with a normal distribution, mean = µ, and standard deviation = σ, then the following holds for any sample size:

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The Standard Error • The standard error is just another name for the standard deviation of the sampling distribution.

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The Central Limit Theorem (Any Distribution) • If a random variable has any distribution with mean = µ and standard deviation = σ, the sampling distribution of x will approach a normal distribution with mean = µ and standard deviation =  n as n increases without limit.

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Sample Size Considerations • For the Central Limit Theorem (CLT) to be applicable: – If the x distribution is symmetric or reasonably symmetric, n ≥ 30 should suffice. – If the x distribution is highly skewed or unusual, even larger sample sizes will be required. – If possible, make a graph to visualize how the sampling distribution is behaving.

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Critical Thinking • Bias – A sample statistic is unbiased if the mean of its sampling distribution equals the value of the parameter being estimated. • Variability – The spread of the sampling distribution indicates the variability of the statistic.

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Sampling Distributions for Proportions pˆ  r n • If np > 5 and nq > 5, then pˆ can be approximated by a normal variable with mean and standard deviation  pˆ  p and pq  pˆ  n

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Continuity Corrections • Since pˆ is discrete, but x is continuous, we have to make a continuity correction. • For small n, the correction is meaningful.

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Control Charts for Proportions • Used to examine an attribute or quality of an observation (rather than a measurement). • We select a fixed sample size, n, at fixed time intervals, and determine the sample proportions at each interval. • We then use the normal approximation of the sample proportion to determine the control limits.

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P-Chart Example

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