11.6 Facts about the correlation coefficient for linear regression  (Page 2/3)

Performing the hypothesis test

Setting up the hypotheses:

• Null Hypothesis: Ho: ρ=0
• Alternate Hypothesis: Ha: ρ≠0

What the hypotheses mean in words:

• Null Hypothesis Ho: The population correlation coefficient IS NOT significantly different from 0. There IS NOT a significant linear relationship(correlation) between x and y in the population.
• Alternate Hypothesis Ha: The population correlation coefficient IS significantly DIFFERENT FROM 0. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

Drawing a conclusion:

• There are two methods to make the decision. Both methods are equivalent and give the same result.
• Method 1: Using the p-value
• Method 2: Using a table of critical values
• In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05
• Note: Using the p-value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

Method 1: using a p-value to make a decision

• The linear regression t-test LinRegTTEST on the TI-83+ or TI-84+ calculators calculates the p-value.
• On the LinRegTTEST input screen, on the line prompt for β or ρ, highlight " ≠ 0 "
• The output screen shows the p-value on the line that reads "p=".
• (Most computer statistical software can calculate the p-value.)

If the p-value is less than the significance level (α = 0.05):

• Decision: REJECT the null hypothesis.
• Conclusion: "The correlation coefficient IS SIGNIFICANT."
• We believe that there IS a significant linear relationship between x and y. because the correlation coefficient is significantly different from 0.

If the p-value is not less than the significance level (α = 0.05)

• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: "The correlation coefficient is NOT significant."
• We believe that there is NOT a significant linear relationship between x and y. because the correlation coefficient is NOT significantly different from 0.

Calculation notes:

• You will use technology to calculate the p-value. The following describe the calculations to compute the test statistics and the p-value:
• The p-value is calculated using a $t$ -distribution with n-2 degrees of freedom.
• The formula for the test statistic is t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}} . The value of the test statistic, $t$ , is shown in the computer or calculator output along with the p-value. The test statistic $t$ has the same sign as the correlation coefficient $r$ .
• The p-value is the probability (area) in both tails further out beyond the values - $t$ and $t$ .
• For the TI-83+ and TI-84+ calculators, the command 2*tcdf(abs(t),10^99, n-2) computes the p-value given by the LinRegTTest; abs(t) denotes absolute value: | $t$ |

Third exam vs final exam example: p value method

• Consider the third exam/final exam example .
• The line of best fit is: $\hat{y}=-173.51+\text{4.83x}$ with $r=0.6631$ and there are n = 11 data points.
• Can the regression line be used for prediction? Given a third exam score ( $x$ value), can we use the line to predict the final exam score (predicted $y$ value)?