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.
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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