12 The distribution of the heights of adults in the United S

12. The distribution of the heights of adults in the United States is bimodal (i.e., has two peaks).

(a) If a sample of 2000 adults were selected, would the sampling distribution of the sample mean be approximately Normal? Why or why not?

(b) If the sample size were increased, would the sampling variability increase, decrease, or stay the same? Explain your answer.

Solution

Solution:

(a)When sample size(n) is large,the central limit theorem states that the distribution of sample means is approximately normal regardless of the shape of the population distribution.

Central Limit Theorem

The central limit theorem states that the sampling distribution of the mean of any independent,random variable will be normal or nearly normal, if the sample size is large enough.

How large is \"large enough\"? The answer depends on two factors.

In practice, some statisticians say that a sample size of 30 is large enough when the population distribution is roughly bell-shaped. Others recommend a sample size of at least 40. But if the original population is distinctly not normal (e.g., is badly skewed, has multiple peaks, and/or has outliers), researchers like the sample size to be even larger.

(b) the standard deviation is a measure of variability.The standard deviation of the sampling distribution is

= /n.

where is standard deviation of population and

n is sample size

sigma/sqrt(n).Thus,as sample size (n) increases, the variabliity decreases.

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