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5.2 The Sampling
Distribution of a Sample
Mean
One thing we’ve been saying (or hoping) is that a sample
should have approximately the same statistical
properties as the population. In particular, we would like
the mean of a sample to be approximately the mean of
our population.
Given a random variable, one thing we can do is to look
at a fixed sample size and write down a probability
distribution of the mean. In this way, the mean itself is a
random variable.
The question is then: how does the probability
distribution of the mean of a fixed sample size relate to
the mean of the population? We would ask the same
question of standard deviation.
Central Limit Theorem
The above questions and others are answered
by the Central Limit Theorem:
Let x represent data values from a population
with mean μ, and let x represent the sample
mean defined for random samples of size n.
Then x is a random variable with the following
properties.
1. The mean of x is μ.
2. The standard deviation of x is σ/n1/2.
3. x will be approximately normal when n is
sufficiently large (n greater than 30).
The last part of the theorem tells us that
even if x is not normal, x is closer and
closer to normal for larger values of n.
This may be offered as one reason why so
many variables in the real world are
approximately normal. If your random
variable can be considered a mean, then it
is approximately normal.
Examples
At a large factory, the mean wage is
$42,500 and the standard deviation is
$2000. What is the probability that the
mean wage of 75 randomly selected
workers will exceed $43,000?
A biology teacher had noted that the scores on a
standard exam attained by students from her
past classes had a mean of 74 and a standard
deviation of 14. The teacher decided to use a
new book. Using the new book, a class of 50
students scored a mean of 78 on the
standardized exam. Find the probability that a
class of 50 students using the new book would
have a sample mean as large or larger than 78 if
the new book were equivalent to the old book.
Is this evidence to support the teacher’s claim
that the new book is superior?
We are now beginning to hint at the idea of
hypothesis testing and p-values.
Practice Problem
The scores of high school seniors on the ACT in 2003
had mean μ=20.8 and σ=4.8. The distributions of scores
is not normal.
1. What is the approximate probability that a single
random student chosen from all those taking the test
scores 23 or higher?
2. What are the mean and standard deviation of the
sample mean score of a SRS of 25 students?
3. What is the approximate probability that the mean
scores of these students is 23 or higher?
4. Which of your two Normal probability calculations is
more accurate? Why?