example of cross-
sectional data.
f: State the central limit theorem and describe its importance.
The central limit theorem tells us that for a population with a mean µ and a finite
variance σ
2
, the sampling distribution of the sample means of all possible samples of size n
will be approximately normally distributed with a mean equal to µ and a variance equal to
σ
2
/n.
The central limit theorem is extremely useful because the normal distribution is relatively easy
to work with when doing hypothesis testing and forming confidence intervals. We can make
very specific inferences about the population mean, using the sample mean, no matter the
distribution of the population, as long as the sample size is "large."
What you need to know for the exam:
1.
If the sample size n is sufficiently large (greater than 30), the sampling distribution of
the sample means will be approximately normal.
2.
The mean of the population, µ, and the mean of all possible sample means, µ
x
, are
equal.
3.
The variance of the distribution of sample means is σ
2
/n.
g: Calculate and interpret the standard error of the sample mean.
Standard error of the sample means is the standard deviation of the sampling distribution
of the sample means. The standard error of the sample means when the standard deviation
of the population is known is calculated by: σ
x
= σ/
√ n, where: σ
x
= the standard error of the
sample means, σ = the standard deviation of the population, and n = the size of the sample.
Example: The mean hourly wage for Iowa farm workers is $13.50 with a standard deviation
of $2.90. Let x be the mean wage per hour for a random sample of Iowa farm workers. Find
the mean and standard error of the sample means, x, for a sample size of 30.
The mean μ
x
of the sampling distribution of x is μ
x
= μ = $13.50. Since σ is known, the
standard error of the sample means is: σ
x
= σ/
√n = 2.90 / √30 = $.53. In conclusion, if
you were to take all possible samples of size 30 from the Iowa farm worker population and
prepare a sampling distribution of the sample means you will get a mean of $13.50 and
standard error of $.53.
h: Distinguish between a point estimate and a confidence interval estimate of a population
parameter.
Point estimates
are single (sample) values used to estimate population parameters. The
formula we use to compute the point estimate is called the estimator. For example, the
sample mean X bar is an estimator of the population mean µ, and is computed using the
following formula:
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