$3.$ The t statistic, the t distribution, and sample size

The average duration of labor from the first contraction to the birth of the baby in women over $35$ who have not previously given birth and who did not use any pharmaceuticals is $16$ hours.

Suppose you have a sample of $35$ women who exercise daily, and who have an average duration of labor of $16\mathrm{.}9$ hours and a sample variance of $39\mathrm{.}7$ hours. You want to test the hypothesis that women who exercise daily have a different duration of labor than all women.

Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is sM

$=.$ The t statistic is $.$

Now suppose you have a larger sample size n $=89.$ Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard error is $.$ The new t statistic is $.$

Note that the t statistic becomes as n becomes larger.

Use the Distributions tool to look at the t distributions for different sample sizes. To do this, choose the Degrees of Freedom for the first sample size on the slider, and click the radio button with the single orange line. Move the orange vertical line to the right until the number below the orange line is located on the t statistic. The probability of getting that t statistic or one more extreme will appear in the bubble with the orange type. Now repeat the process for the other sample.

What is the probability of getting the t statistic or something more extreme for the sample size of n $=35?$ p $=.$ What is the probability of getting the t statistic or something more extreme for the sample size of n $=89?$ p $=.$

The t distribution is with a smaller n$.($Hint: To best see this, click the radio button in the tool with no vertical lines. Slowly move the Degrees of Freedom slider from the smallest value to the largest value, and observe how the shape of the distribution