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What distribution does the test statistic follow?
The test statistic follows the t distribution with n-1 degrees of freedom. The test statistic z is used to compute the P-value for the t distribution, the probability that a value at least as extreme as the test statistic would be observed under the null hypothesis.
How is the distribution of sample means constructed?
To create a sampling distribution a research must (1) select a random sample of a specific size (N) from a population, (2) calculate the chosen statistic for this sample (e.g. mean), (3) plot this statistic on a frequency distribution, and (4) repeat these steps an infinite number of times.
When to use the t distribution in statistics?
We will see that the t distribution is a helpful substitute for the normal distribution when we model a sample mean x ¯ that comes from a small sample. While we emphasize the use of the t distribution for small samples, this distribution may also be used for means from large samples.
How did the Student t distribution get its name?
The name comes from the fact that Gosset wrote under the pen name “Student.” Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and used the Student’s t-distribution only for sample sizes of at most 30.
Why does the Student’s t distribution have more probability in its tails?
The Student’s t-distribution has more probability in its tails than the standard normal distribution because the spread of the t-distribution is greater than the spread of the standard normal. So the graph of the Student’s t-distribution will be thicker in the tails and shorter in the center than the graph of the standard normal distribution.
How does the shape of the Student’s t distribution change?
The exact shape of the Student’s t-distribution depends on the degrees of freedom. As the degrees of freedom increases, the graph of Student’s t-distribution becomes more like the graph of the standard normal distribution.