Chapter 1. The Central Limit Theorem

Introduction

Statistical Applets

Choose a population distribution (Exponential, Uniform, or Normal) and a sample size, then click the button to generate 10,000 samples and plot the distribution of sample means. Click "Show Normal curve" to compare this distribution with the Normal curve predicted by the Central Limit Theorem.

Click the "Quiz Me" button to complete the activity.

The Central Limit Theorem says that the distribution of sample means of n observations from any population with finite variance gets closer and closer to a Normal distribution as n increases. More specifically, for a population of individual observations with mean μ and standard deviation σ, the Central Limit Threorem says that the means of samples of size n drawn from this population will approximate a Normal distribution whose mean is also μ and whose standard deviation is .

This applet illustrates the Central Limit Theorem by allowing you to generate thousands of samples with various sizes n from a exponential, uniform, or Normal population distribution. You can then compare the distribution of sample means against the Normal distribution with the standard deviation predicted by the Central Limit Theorem.

Question 1.1

When n = 2, the sampling distribution for the population with the exponential distribution is q8oEs4xwQRwqRTIBQ9YrqHRSmK6YsOST61AmFtD0PGq1ghvs3dDNoslDtyXxpkpF, whereas when n = 50, the sampling distribution for the exponential population is mc7NoJWraBU+Cm/gMHsqD/E23aLhODSejQAN03Cz2nXzfm/YXcD/FyepmZiwM3Tl.

2
Try again.
Incorrect. See above for the correct answers.
Great job.

Question 1.2

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
2
Correct.
Try again.
Incorrect.

Question 1.3

29qPpJBluEGiyLsHhpd3GBeg3xInkFFuMkxenuv8But409MMoKPg3HHifplTk8cdk8SGsEVlClun8wx3vNwvHpz1lsNQPy0QXwghFY/NXi65y6Np6olcztjHu08OLnPt6Nm/8RotjAMka8j9JvyPs/nER2QaJmVOCM/n5mL2hyU7NXFIn3eDndJGveiUUsDWI59w+BcP0DBQkk6hF8G+qpSMIs1uvf4p
The sampling distribution for the exponential population distribution might still be very slightly skewed to the left, but other than this the shapes of the three distributions are virtually identical Normal-shaped curves.