Central Limit Theorem
Chapter 14
Supplemental Instruction
Iowa State University / Leader: / Carly
Course: / Stat 226
Instructor: / (Various)
Date: / 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) / Probability that X = x
1
2
3
4
5
6

Exercise Questions:

Recall the premise of the Central Limit Theorem: the mean of a random sample (X) will approximately follow a normal distribution with mean μ and standard error σn, regardless of the 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.

3.) How does the CLT come into play here?

4.) If the population has a standard deviation of σ and the mean of a sample distribution has a standard error of σn , 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?

5.) Looking at the each histogram, which one appears to have the least variance?

Self Quiz:

1.)  The Central Limit Theorem says that the sampling distribution of the sample mean is approximately normal if…

  1. All possible samples are selected
  2. The sample size is large
  3. 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…

  1. Equal to the population mean divided by the square root of the sample size
  2. Close to the population mean if the sample size is large
  3. 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…

  1. Equal to the population standard deviation divided by the square root of the sample size
  2. Close to the population standard deviation if the sample size is large
  3. 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…

  1. 7.5
  2. 8
  3. 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…

  1. 0.3
  2. 1.5
  3. 7.5