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Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 7 | 1
Chapter 7 Introduction to Sampling Distributions
• Sampling Distributions• The Central Limit Theorem• Sampling Distributions for Proportions
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7.1 Sampling Distributions
• StatisticA statistic is a numerical descriptive measure of a
sample.• ParameterA parameter is a numerical descriptive measure of
a population.
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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. In such cases, we will use a statistic to make inferences about a corresponding population parameter. The followings are the principal types of inferences.
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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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Examples
• Example 1/p295.
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7.2 The Central Limit Theorem
• 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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Example
Suppose a team of biologists has been studying the Pinedale children’s fishing pond. Let x represent the length of a single trout taken at random from the pond.This group of biologists has determined that x has a normal distribution withmean m10.2 inches and standard deviation s1.4 inches.(a) What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?
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Example
b) What is the probability that the mean length of five trout taken at random is between 8 and 12 inches?
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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 will approach a normal distribution with mean = µ and standard deviation = as n increases without limit.
x
n
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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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n
rp ˆ
p̂
pp ˆ
n
pqp ˆ
• If np > 5 and nq > 5, then can be approximated by a normal variable with mean and standard deviation and
7.3 Sampling Distributions for Proportions
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Continuity Corrections
• Since is discrete, but x is continuous, we have to make a continuity correction.
• For small n, the correction is meaningful.
p̂
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Examples
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ExamplesThe annual crime rate in the Capital Hill neighborhood of Denver is 111 victims per 1000 residents. This means that 111 out of 1000 residents have been the victim of at least one crime (Source:Neighborhood Facts, Piton Foundation). These crimes range from relatively minor crimes (stolen hubcaps or purse snatching) to major crimes (murder). The Arms is an apartment building in this neighborhood that has 50 year-round residents.Suppose we view each of the n=50 residents as a binomial trial. The random variable r (which takes on values 0, 1, 2, . . . , 50) represents the number of victims of at least one crime in the next year.
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Example
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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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Procedure
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(c) Signal III: at least two out of three consecutivepoints beyond a control limit (on the same side).If no out-of-control signals occur, we say that the process is “in control,”while keeping a watchful eye on what occurs next.
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Example
Anatomy and Physiology is taught each semester. The course is required for several popular health-science majors, so it always fills up to its maximum of 60 students. The dean of the college asked the biology department to make a control chart for the proportion of A’s given in the course each semester for the past 14 semesters. Using information from the registrar’s office, the following data were obtained. Make a control chart and interpret the result.
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Example
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Example
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P-Chart Example
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Example
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Example
(e) Conclusion: The biology department can tell the dean that the proportion of A’s given in Anatomy and Physiology is in statistical control, with the exception of one unusually good class two semesters ago.
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