MA121 Study Guide


Unit 4: Estimation with Confidence Intervals

4a. Explain the central limit theorem, and use it to construct confidence intervals

  • What role does the Central Limit Theorem have for estimation and sampling distributions?

We discussed the Central Limit Theorem in Unit 3. It is CRITICAL to note that if the distribution is non-normal, we MUST still have a large sample size. In other words, the Central Limit Theorem still applies. If it does not (non-normal distribution AND low sample size) then neither Z nor T will work.

The Central Limit Theorem still requires a normally distributed population or a sample size above 30. Using the T distribution instead of Z does not absolve us of these requirements.

Review the Central Limit Theorem in: 


4b. Compare t-distributions and normal distributions

  1. What is the difference between a normal distribution and a T distribution?
  2. Where would you need to use a T, instead of a normal (X), or the standard normal (Z) distributions?

Refer to learning objective 2k to review the progression from continuous distribution, to symmetric, to bell-shaped, to normal (X) and standard normal (Z).

Now, let's introduce the T distribution, which you can think of as a "brother" to the normal distribution in this hierarchy.

The Student's T distribution is similar to the normal distribution except that it is slightly shorter and flatter, with heavier tails than X or Z. In other words, the area to the right of two standard deviations in a normal distribution is 0.025. In a T distribution it will be larger. The T family of distributions is defined by a single parameter called the degrees of freedom which your book symbolizes simply as df although some texts will use other symbols like \nu. The degrees of freedom, when we are talking about the T distribution is equal to the sample size, minus one.

We use the T distribution in the same situations above as we use the Z, however we must use T instead when we are unsure of the shape of the population distribution, or we are using an estimate of the standard distribution (s instead of σ).

To find an area that is bound by the T distribution, we can use technology similar to what we use for the Z distribution. There is a T distribution table with one row each for most common values of T. The drawback is that we cannot use this to find the exact area, only the "T score" given a particular area.

Note that with a T distribution, as the sample size (thus the degrees of freedom) increases, it is nearly indistinguishable from the Z distribution. In fact, the Z distribution IS the T distribution with an "infinite" degrees of freedom! Using the T distribution gives us a larger margin of error to compensate for the fact that the exact standard deviation of the population is not known.

Review this material in T Distribution.


4c. Apply and interpret the central limit theorem for sample averages

  • We saw earlier that the Central Limit Theorem holds if we have either a large sample size OR an underlying normal distribution. Does this affect whether we should use a Z or T distribution for estimation, or whether we can use either at all?

Whenever we perform an estimation, we must use a T distribution for a small sample size or an unknown value of the population standard distribution. The formula for using T distribution to find the margin of error and confidence intervals is the same as the one for Z, except we substitute σ with s. The difference is that instead of finding the cutoff value for the z score in the formula, we have to find the t score instead. This introduces an extra parameter. In addition to using the alpha value for the area right of the tail, we need to use n−1 (one less than the sample size) for the parameter "df". This will result in a larger margin of error because, as we stated earlier, T distributions have a heavier tail. 

Review this material in The Sampling Distribution of the Sample Mean.


4d. Calculate, describe, and interpret confidence intervals for population averages and one population proportions

  • What does the confidence interval for a data sample tell you?
  • Why was it necessary for you to learn about sampling distributions first?

We use inferential statistics to interpret samples and make conclusions about the population of data. The general method for making these interpretations is roughly the same, whether it is for population averages, proportions, averages of two different populations, standard deviations, or any other statistic.

First, we find a point estimate which is usually the sample mean or proportion. Then, given the level of confidence desired, the sample size, and the standard deviation of the population, we can find a margin of error and subtract or add it to the point estimate to get a confidence interval for the population parameter.

When we refer to level of confidence we mean the likelihood that our confidence interval contains the true population mean or proportion (or other parameter). Common confidence levels are 90%, 95%, and 99%. A higher confidence level gives a higher likelihood that the interval contains the population parameter. However the price to be paid is that naturally a higher confidence interval will be wider.

So for example, we could say there is a 95 percent probability that the population mean is between 10 ± 2.8, [7.2, 10.8]. If we want to be 99 percent confident, we have to increase the width of the interval, perhaps 10 ± 3.2. A 100 percent confidence interval is not possible since that would require an infinitely wide margin of error. So there is a give and take. Choosing a confidence level can be more art than science. Balance your desire for accuracy with the need to keep the margin of error low.

The reason you had to learn about sampling distributions first is because inferential statistics involve predicting the characteristics of a population based on a sample. In order to do so, we must first study how those samples behave.

Use the given formulas to calculate the margin of error for the population mean, given a large population (this is when you use the standard normal Z distribution). If the population is small, or it is large and the standard deviation of the population is unknown, you must use the Student's T.

The formulas are very similar, except for the distribution. You would use the inverse Z distribution or inverse T distribution with given degrees of freedom. You also have to plug in the Z or T score corresponding to ∝/2, where ∝ is equal to the tail area on one side. If you, for example, are trying to find a 90 percent confidence interval, that will leave 10 percent area on the tails, which you divide in half to come up with ∝/2 = .10/2 = 0.05.

Review this material in:


4e. Interpret the student-t probability distribution as the sample size changes

  • How is the student-t distribution related to sample size? Why does this not matter for normal or standard normal distributions?
  • What happens to the student t distribution as the sample size increases? What distribution does it begin to resemble?

The student-t distribution, like the normal distribution, is a family of distributions defined by one or more parameters. A normal distribution is defined by its mean and standard deviation. The student-t distribution is defined by the number of degrees of freedom, which is equal to the sample size minus 1. 

The larger the sample size, the lower the margin of error, and the larger degrees of freedom, which combined will give you a smaller margin of error. The larger the degrees of freedom, the more the T distribution resembles the standard normal (Z) distribution. In fact, a Z distribution IS by definition the same as a T distribution with infinite degrees freedom!

Review this material in:


Unit 4 Vocabulary 

  • Central Limit Theorem
  • Confidence interval
  • Degrees of freedom
  • Level of confidence
  • Margin of error
  • Normal distribution
  • Point estimate
  • Standard normal distribution
  • Student's T distribution