## Theory-Based Approach

Because the null distribution of sample slopes is approximately normal, we can standardize the sample slope into a *t*-statistic and use the *t*-distribution to estimate the p-value. (There are of course *technical conditions* to check for this, but we aren't going to worry about them too much this time.)

In the applet, above the null distribution use the Statistic pull-down menu to select the **t****-statistic **optio . Check the new box to **Overlay t-distribution**.

(o) Enter the test statistic you calculated in (m) and press **Count**. (Don't worry if the applet complains about your value, could be rounding differences.) Does the theoretical p-value provide a reasonable approximation to the simulated p-value?

Now check the box under the original scatterplot to display the **Regression Table**. Click here for an explanation of the regression output. Take a **screen capture** of this output (null distribution and regression table) and copy it into your report.

(p) How does the SE value corresponding to the *friends *row compare to the standard deviation of the slopes found in (j)? (**Note**: They may not be all that close because they are actually based on slightly different assumptions, but they do both aim to estimate the same thing.)

(q) How does this *t*-value compare to what you found in (m)?

(r) How does the p-value in the output for the Regression Table compare to the one-sided p-value you estimated in (k)?

**Note: **The *t* value in the *friends* row is our test statistic and the p-value reported there is actually always the *two-sided* p-value for testing H_{0}: = 0 vs. H_{a}: 0. When you have a one-sided alternative hypothesis, you can take the p-value reported by in the regression output and cut it in half to determine the one-sided p-value.

(s) Check the box underneath the Regression Table to display the theory-based 95% Confidence interval for slope. How does it compare to the interval you found in (n)? Write a one-sentence interpretation of this interval.