Central Limit Theorem and Hypothesis Testing

Central Limit Theorem and Hypothesis Testing

Author: Al Greene

This learning packet should review:

• New terms and definitions
• Review confidence intervals, hypothesis tests and a normal distribution
• Understand and explain the Student’s t distribution, history and usage and when to use the Student’s t methods
• Define and explain the Central Limit Theorem and important uses of the theorem

This packet introduces the central limit theorem and Student's t distribution, two very important concepts in statistics. It also relates them to terms you have already seen, such as confidence intervals and hypothesis testing.

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New Terms

In this packet, we will be introduced to the following terms:

  • Student's t distribution
  • Central Limit Theorem
  • Analysis of Variance

Some terms that should already be known are:

  • Hypothesis Testing
  • Confidence Interval

Source: Greene

Central Limit Theorem and Hypothesis Testing

This video discusses the central limit theorem, student's t distribution, how they relate to confidence intervals and hypothesis testing, and introduces the idea of Analysis of Variance (ANOVA).

Source: Greene

Central Limit Theorem

This shows a fun demonstration on how the central limit theorem works.

Source: YouTube

Real Life Central Limit Theorem Examples

Let's look at the random variable X = year on a penny. It is pretty safe to assume that this will not be normally distributed. There will be plenty of old pennies, plenty of new pennies, and plenty inbetween. But, if we take a large random sample of pennies, calculate the average year of all those pennies, repeat this many times, and graph our results, the distribution will now appear normal! This is the beauty of the central limit theorem. Regardless of the underlying distribution of the original population, the new sampling distribution will be normal as long as you take large enough samples.

Another tangible example of the central limit theorem would be the length of words in a book. This is not a normal distribution either; it is in fact right skewed. The shortest words will be 1 letter (a, I) there will be a peak around 4 letters, and it can go as high as 10, 12, or more. But the central limit theorem applies to this situation as well. If we take a large random sample of words, find the average length, repeat this many times, and graph the results, the distribution will now be normal.

Can you come up with your own example of a distribution that the central limit theorem can be applied to?

Source: Greene