# 9.4.1 - Hypothesis Testing for the Population Correlation

9.4.1 - Hypothesis Testing for the Population Correlation

In this section, we present the test for the population correlation using a test statistic based on the sample correlation.

Assumptions

As with all hypothesis test, there are underlying assumptions. The assumptions for the test for correlation are:

• The are no outliers in either of the two quantitative variables.
• The two variables should follow a normal distribution
Hypotheses

If there is no linear relationship in the population, then the population correlation would be equal to zero.

$H_0\colon \rho=0$ ($X$ and $Y$ are linearly independent, or X and Y have no linear relationship)

$H_a\colon \rho\ne0$ ($X$ and $Y$ are linearly dependent)

Research Question

Is there a linear relationship?

Is there a positive linear relationship?

Is there a negative linear relationship?

Null Hypothesis

$\rho=0$

$\rho=0$

$\rho=0$

Alternative Hypothesis

$\rho\ne0$

$\rho>0$

$\rho<0$

Type of Test

Two-tailed, non-directional

Right-tailed, directional

Left-tailed, directional

Test Statistic

Under the null hypothesis and with above assumptions, the test statistic, $t^*$, found by:

$t^*=\dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

which follows a $t$-distribution with $n-2$ degrees of freedom.

As mentioned before, we will use Minitab for the calculations. The output from Minitab previously used to find the sample correlation also provides a p-value. This p-value is for the two-sided test. If the alternative is one-sided, the p-value from the output needs to be adjusted.

## Example 9-7: Student height and weight (Tests for $\rho$)

For the height and weight example (university_ht_wt.TXT), conduct a test for correlation with a significance level of 5%.

The output from Minitab is:

#### Correlation: height, weight

##### Correlations
Pearson correlation

P-value

0.711

0.000

For the sake of this example, we will find the test statistic and the p-value rather than just using the Minitab output. There are 28 observations.

The test statistic is:

\begin{align} t^*&=\dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}}\\&=\dfrac{(0.711)\sqrt{28-2}}{\sqrt{1-0.711^2}}\\&=5.1556 \end{align}

Next, we need to find the p-value. The p-value for the two-sided test is:

$\text{p-value}=2P(T>5.1556)<0.0001$

Therefore, for any reasonable $\alpha$ level, we can reject the hypothesis that the population correlation coefficient is 0 and conclude that it is nonzero. There is evidence at the 5% level that Height and Weight are linearly dependent.

## Try it!

For the sales and advertising example, conduct a test for correlation with a significance level of 5% with Minitab.

Sales units are in thousands of dollars, and advertising units are in hundreds of dollars.

Sales (Y) Advertising (X)
1 1
1 2
2 3
2 4
4 5
The Minitab output gives:

#### Correlation: Y,X

##### Correlations
Pearson correlation

P-value

0.904

0.035

The sample correlation is 0.904. This value indicates a strong positive linear relationship between sales and advertising.

For the Sales (Y) and Advertising (X) data, the test statistic is...

$t^*=\dfrac{(0.904)\sqrt{5-2}}{\sqrt{1-(0.904)^2}}=3.66$

...with df of 3, we arrive at a p-value = 0.035. For $\alpha=0.05$, we can reject the hypothesis that the population correlation coefficient is 0 and conclude that it is nonzero, i.e., conclude that sales and advertising are linearly dependent.

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