# 4.2 Testing the significance of the correlation coefficient  (Page 2/6)

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• Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. 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 zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

## Drawing a conclusion:

There are two methods of making the decision. The two 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

To calculate the p -value using LinRegTTEST:
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: "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 zero."

## 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 zero."

## Calculation notes:

• You will use technology to calculate the p -value. The following describes 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 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: ŷ = -173.51 + 4.83 x 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)?

H 0 : ρ = 0

H a : ρ ≠ 0

α = 0.05

• The p -value is 0.026 (from LinRegTTest on your calculator or from computer software).
• The p -value, 0.026, is less than the significance level of α = 0.05.
• Decision: Reject the Null Hypothesis H 0
• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

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