Student’s t distribution

Consider a sampling situation where the population has a normal distribution with unknown . Because is unknown, an intuitive approach is to estimate by the sample standard . Just as we did in the large sample situation, we consider the ratio

This random variable does not follow a standard normal distribution. Its distribution is known as Student’s t distribution.

The graph of the t -distribution resembles the graph of the standard normal distribution: they both are symmetric, bell shaped curves with mean equal to zero. The graph of the Student’s t distribution is lower at the center and higher at the extremities than the standard normal curve. (Fig. 6.4).

The new notation t is required in order to distinguish it from the standard normal variable Z. As the number of degrees of freedom increases, the difference between t distribution and the standard normal distribution becomes smaller and smaller.

The qualification “with (n -1) degrees of freedom” is necessary, because with each different sample size or value of (n -1), there is a different t distribution.

Definition:

The number of degrees of freedom is defined as the number of observations that can be chosen freely.

Example:

Suppose we know that the mean number of 5 values is 25. Consequently, the sum of these 5 values is 125 . Now how many values out of 5 can be chosen freely so that the sum of these 5 values is 125? The answer is that we can freely choose 5-1=4 values. Suppose we choose 15, 35, 45, and 10 as the 4 values. Given these 4 values and the information that the mean of the 5 values is 25, the value is

Thus, once we have chosen 4 values, the fifth value is automatically determined. Consequently, the number of degrees of freedom for this example is

We subtract 1 from n because we lose one degree of freedom to calculate the mean.

The t - table in the Appendix (see table 4) is arranged to give the value t for several frequently used values of and for a number of values (n -1).

Definition:

A random variable having the standard distribution with (Greek letter nu)

Degrees of freedom will be denoted by (Fig. 6.8). Then is defined as the number for which

Example: Find

Solution:

In words it means we need to find a number that is exceeded with the probability 0.10 by a Student’s t random variable with 5 degrees of freedom.

From table 4 of the Appendix we read that . (Fig. 6.9).

Similarly, to for Student’s t distribution the value is defined as

.

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