Download 8.1.4 The Central Limit Theorem - University of Northern Colorado

Distribution of the Sample Mean
Page 1
The Central Limit Theorem
The goal of this activity is to apply the Central Limit Theorem.
The Central Limit Theorem
Suppose that a simple random sample of size n is obtained from a large population with mean
µ and standard deviation σ. The sampling distribution of x will have the following
characteristics:
1. As the sample size increases, the sampling distribution will approach a normal
distribution.
2. The mean of sample means is:
µx = µ.
3. The standard deviation of the sample mean (called the standard error of the mean) is:
σ
σx =
n
1. Nägele’s Rule (developed in the 1850’s) says the gestation period of humans is normally
distributed with a mean of 266 days and a standard deviation of 16 days. A medical research
team plans to study 10 pregnant women. The researchers want to compare their results with
other such studies.
a. Find the mean and standard deviation expected for comparison.
b. Graph the population distribution and the sampling distribution in the same viewing
window on your graphing calculator. Compare the two distributions. The following
distribution function on your calculator may help. (Hint: use X as the value.)
normalpdf(value,mean,standard deviation)
c. Determine the probability that the women in the study will have a mean gestation period
of 260 days or less.
Robert A. Powers
University of Northern Colorado
Distribution of the Sample Mean
Page 2
Practical Guidelines for the Central Limit Theorem
If a random variable X is normally distributed, then the sampling distribution of x is normally
distributed.
If a random variable X is not normally distributed, but n > 30, then the sampling distribution of
x is approximated reasonably well by a normally distribution, except under relatively rare
circumstances.
2. Market analysts use a collection of stocks to determine trends. For example, the S & P 500 is
a collection of 500 stocks of publicly traded companies whose values are used to determine
monthly rates of return. A statistician thinks this is inefficient and proposes a new method
called the R A P 20. This method starts with the current market value of stocks. Then
randomly selects 20 stocks of publicly traded companies to determine their change in market
value for the day and adjusts the market value of stocks accor
The Central Limit Theorem for a Finite Population
Suppose that a simple random sample of size n is obtained from a finite population of size N
with mean µ and standard deviation σ. The sampling distribution of x will have the following
characteristics:
1. As the sample size increases, the sampling distribution will approach a normal
distribution.
2. The mean of sample means is:
µx = µ.
3. The standard deviation of the sample mean is:
N −n σ
σx =
⋅
N −1 n
Robert A. Powers
University of Northern Colorado