Central Limit Theorem
Chapter 14
Supplemental Instruction
Iowa State University
Leader:
Course:
Instructor:
Date:
Carly
Stat 226
(Various)
3/6/13
Agenda:
Opening Activity: Last-Minute Homework Questions
Die Rolling Activity
Closing Activity: Self Quiz
Die Rolling Activity:
Let X be a random variable representing the roll of a fair 6-sided die. Complete the
following table which will represent the theoretical distribution of X (Value of the Die and its
corresponding probability).
Table 1:
Die Value (x)
1
2
3
4
5
6
Probability that X = x
1/6
1/6
1/6
1/6
1/6
1/6
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Population Distribution
0.20
0.18
0.16
Probability
0.14
0.12
0.10
1
2
3
4
5
6
Histogram for n = 2
Frequency
40
30
20
10
Series1
0
1
2
3
4
5
6
Bin
Histogram for n = 10
Frequency
80
60
40
20
Frequency
0
1
2
3
4
5
6
More
Bin
Frequency
Histogram for n = 30
100
80
60
40
20
0
Frequency
1
2
3
4
Bin
5
6
More
Exercise Questions:
Recall the premise of the Central Limit Theorem: the mean of a random sample (𝑋̅) will
𝜎
approximately follow a normal distribution with mean μ and standard error , regardless of the
√𝑛
distribution of the population (X). The theory requires a sample size of at least 30 if the
population distribution is unknown.
1.) Label the histograms on the previous page for n = {2, 10, 30}.
2.) Comment on the difference in shape between each of the histograms, including the population
distribution.
As the sample size increases, we see that the spread of the sampling distribution becomes
narrower, more normal, and more symmetric.
3.) How does the CLT come into play here?
The distribution of the population is symmetric and rectangular – not normal. For sufficiently
large sample sizes, the population distribution appears normal.
4.) If the population has a standard deviation of σ and the sample distribution has a standard error
𝜎
of 𝑛 , which one of these two values will be lower if n>1? Hence, which one of these two
√
distributions do you expect to have lower variability: the population distribution or sampling
distribution?
The standard error of the sampling distribution decreases as the sample size increases. We
would expect the sampling distribution to have a lower variability if n is greater than 1.
5.) Looking at each histogram, which one appears to have the least variance?
The last one – the histogram for n=30.
Self Quiz:
1.) The Central Limit Theorem says that the sampling distribution of the sample mean is
approximately normal if…
a. All possible samples are selected
b. The sample size is large
c. The standard error of the sampling distribution is small
2.) The Central Limit Theorem says that the mean of the sampling distribution of the sample
means is…
a. Equal to the population mean divided by the square root of the sample size
b. Close to the population mean if the sample size is large
c. Exactly equal to the population mean
3.) The Central Limit Theorem says that the standard deviation of the sampling distribution
of the sample mean is…
a. Equal to the population standard deviation divided by the square root of the
sample size
b. Close to the population standard deviation if the sample size is large
c. Exactly equal to the standard deviation of the population distribution
4.) Samples of size 25 are selected from a population with mean 40 and standard deviation
7.5. The mean of the sampling distribution of sample means is…
a. 7.5
b. 8
c. 40
5.) Samples of size 25 are selected from a population with mean 40 and standard deviation
7.5. The standard error of the sampling distribution of sample means is…
a. 0.3
b. 1.5
c. 7.5