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The Law of Large Numbers and the Central Limit Theorem Study Guide (page 2)

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Updated on Oct 5, 2011

Example

Based on a random sample of 4,252 men for the 1988 National Survey of Families and Households, it was reported that men spent a mean of 18.1 hours per week doing housework. Suppose the standard deviation was known to be 12.9 hours.

  1. Assuming a normal distribution and using the information given, sketch the approximate distribution of the number of hours a randomly selected man spent doing housework in 1988.
  2. Based on the graph in the previous question, explain why the population distribution is very unlikely to be normal and why it is most likely skewed to the right.
  3. The researcher believes that 18.1 hours per week is close to the mean time men spent doing housework in 1988. Provide a justification for this reasoning.
  4. What is the approximate sampling distribution of the mean number of hours men spent on housework in 1988 based on samples of size 4,252?

Solution

  1. Examine the graph in Figure 13.3.

     

    Figure13.3

  2. For a normal distribution, 95% of the population is within two standard deviations of the mean. If we assume for the moment that the sample mean is a good estimate of the population mean, 95% of the men would spend between – 7.7 and + 43.9 hours on housework. It is impossible for anyone to work a negative number of hours; the minimum number of hours that could be recorded is zero. Given the large sample size, it is unlikely that the sample mean is such a poor estimate that all values would be nonnegative if we had the true population mean. Thus, the normal is not a reasonable model for this distribution. To have only nonnegative population values with a mean and standard deviation similar to those observed, the population distribution must be skewed to the right.
  3. Because the sample size is large, the sample mean should be close to the population mean by the Law of Large Numbers.
  4. The sample size of 4,252 is certainly large enough to invoke the Central Limit Theorem (n ≥ 30). Therefore, the sample mean has an approximate normal distribution with an estimated mean of 18.1 hours and a standard deviation of .

Example

A female high school student decided that she wanted to date only males who went to her school and had brown eyes. To estimate what proportion of the males in her school would meet these criteria, she randomly selected 20 males from the school. Of these, 12 had brown eyes. Explain why the Central Limit Theorem does not apply.

Solution

The proportion of males in the school with brown eyes is estimated to be = = 0.60. To invoke the Central Limit Theorem, we must have ≥ 10 and n(1 – ) ≥ 10. Here, = 20 × 0.60 = 12 > 10,but n(1 – ) = 2011 – 0.62 = 8 > Only one of the two necessary conditions is met, so it would not be appropriate to apply the Central Limit Theorem here.

The Law of Large Numbers and the Central Limit Theorem In Short

The Law of Large Numbers assures us that the sample mean is getting closer to the population mean as the sample size increases. As long as a large enough sample is taken from a distribution with mean μ and standard deviation σ, the Central Limit Theorem assures us that has an approximate normal distribution with mean μ and standard deviation . We will repeatedly use this fact to draw inferences about populations.

Find practice problems and solutions for these concepts at The Law of Large Numbers and the Central Limit Theorem Practice Exercises.

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