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

Performing the hypothesis test

Setting up the hypotheses:

• Null Hypothesis: $H_o$ : $\rho$ = 0
• Alternate Hypothesis: $H_a$ : $\rho$ ≠ 0

What the hypotheses mean in words:

• Null Hypothesis $H_o$ : 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 $H_a$ : 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%, $\alpha$ = 0.05
• Note: Using the p-value method, you could choose any appropriate significance level you want; you are not limited to using $\alpha$ = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, $\alpha$ = 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 $\beta$ or $\rho$ , 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: "There is sufficient evidence to conclude 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: "There is insufficient evidence to conclude that there is 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 $\mathrm{n-2}$ 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 combined area in both tails.
• An alternative way to calculate the p-value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

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 $\mathrm{n\; =\; 11}$ 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)?