Student's t-Distribution

The t-distribution describes the standardized distances of sample means to the population mean when the population standard deviation is not known, and the observations come from a normally distributed population.

The t-distribution is similar to a normal distribution. Like the normal distribution, the t-distribution is symmetric, and has a smooth shape. Like a standard normal distribution (or z-distribution), the t-distribution has a mean of zero.

Differences:

• The normal distribution assumes that the population standard deviation is known. The t-distribution does not make this assumption. It is based on the sample standard deviation.
• The t-distribution is defined by the degrees of freedom.
• The t-distribution is most useful for small sample sizes, when the population standard deviation is not known, or both. As the sample size increases, the t-distribution becomes more similar to a normal distribution.

In the above graph, all of the distributions have a smooth shape. All are symmetric. All have a mean of zero.

Degrees of Freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample. The shape of the t-distribution depends on the degrees of freedom, which is n - 1. The curves with more degrees of freedom are taller and have thinner tails. All three t-distributions have "heavier tails" than the z-distribution.

The t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. These topics are covered in the reading of "Hypothesis Testing".

Chi-Square Distribution

The Chi-square distribution is a family of distributions. Each distribution is defined by the degrees of freedom.

The figure below shows three different Chi-square distributions with different degrees of freedom.

You can see that the blue curve with 8 degrees of freedom is somewhat similar to a normal curve (the familiar bell curve). But, it has a longer tail to the right than a normal distribution and is not symmetric. Compare the blue curve to the orange curve with 4 degrees of freedom. The orange curve is very different from a normal curve. The purple curve has 3 degrees of freedom and looks even less like a normal curve than the other two curves.

The higher the degrees of freedom for a Chi-square distribution, the more it looks like a normal distribution.

F Distribution

F distribution is used for one-way ANOVA and the test of two variances. Here are some facts about the F distribution.

• The curve is not symmetrical but skewed to the right.
• There is a different curve for each set of dfs.
• The F statistic is greater than or equal to zero.
• As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.

Practice Question 1

A t-test is ______.

I. sensitive to violations of its assumptions
II. defined by the numerator and denominator degrees of freedom
III. a hypothesis test using a statistic (t-statistic) that follows a t-distribution
IV. a hypothesis test concerning a parameter

Correct Answer: III

Practice Question 2

The t-test is used when ______

I. we have large samples and do not know the population variance.
II. we have small samples and know the population variance.
III. we have small samples, do not know the population variance, and the population is normally distributed.
IV. we have large samples and know the population variance.

Correct Answer: I and III

Practice Question 3

Which of the following statements is incorrect?

A. The chi-square distribution is symmetrical just like the normal distribution.
B. Chi-square random variables can only be positive.
C. The chi-square distribution is defined by a single parameter.

Correct Answer: A

The statement that the chi-square distribution is symmetrical just like the normal distribution is incorrect. The chi-square distribution is not symmetrical; it is skewed. All other statements are correct.

Practice Question 4

Which of the following statements is correct?

A. Just like the chi-square distribution, the F-distribution is skewed.
B. The F-distribution is defined by three parameters.
C. The two samples used in the F-test must have the same size.

Correct Answer: A

Just like the chi-square distribution, the F-distribution is skewed. This statement is correct; all other statements are incorrect. The F-distribution is defined by two parameters: the numerator and the denominator degrees of freedom. The sample size of the two samples does not have to be equal.

Practice Question 5

The F-test is used to ______.

A. carry out a hypothesis test for an analysis of variance
B. predict the variance when the sample size is unknown
C. carry out a hypothesis test for an analysis of the standard deviation
D. predict the variance when the sample size is greater than 30

Correct Answer: A

The F-test is used to carry out a hypothesis test for an analysis of variance.

Practice Question 6

Compared to the normal distribution, the Student's t-distribution most likely ______.

A. has fatter tails
B. is more peaked
C. has greater degrees of freedom

Correct Answer: A

The Student's t-distribution has fatter tails and is less peaked compared to the normal distribution.

Practice Question 7

Compared to the normal distribution, the Student's t-distribution most likely ______.

A. has fatter tails
B. is more peaked
C. has greater degrees of freedom

Correct Answer: A

The Student's t-distribution has fatter tails and is less peaked compared to the normal distribution.

Practice Question 8

Which statement is FALSE regarding the t distribution?

A. It is symmetric.
B. It has a bell shape.
C. It has lighter tails than a normal distribution.

Correct Answer: C

The t-distribution is symmetric and bell-shaped, like the normal distribution. However, the t-distribution has heavier tails, meaning that it is more prone to producing values that fall far from its mean.