## Testing Correlation Coefficient

Let $(X_i, Y_i), i=1,2,\cdots, n$ be a random sample of $n$ pairs of observations drawn from a bivariate population with correlation coeffficient $\rho$.

Let $r$ be the observed correlation coefficient between $X$ and $Y$.

We wish to test the hypothesis $H_0 : \rho =\rho_0$ against $H_a : \rho \neq \rho_0$.

## Assumptions

• The population from which, the samples drawn, is a bivariate normal.
• The relationship between $X$ and $Y$ is linear.

## Step by step procedure

The step by step procedure for testing $H_0: \rho = \rho_0$ is as follows:

### Step 1 State the hypothesis testing problem

The hypothesis testing problem can be structured in any one of the three situations as follows:

Situation Hypothesis Testing Problem
Situation A: $H_0: \rho=\rho_0$ against $H_a : \rho < \rho_0$ (Left-tailed)
Situation B: $H_0: \rho=\rho_0$ against $H_a : \rho > \rho_0$ (Right-tailed)
Situation C: $H_0: \rho=\rho_0$ against $H_a : \rho \neq \rho_0$ (Two-tailed)

### Step 2 Define the test statistic

The test statistic for testing above hypothesis is \begin{aligned} Z&=\dfrac{U-\xi}{\sqrt{\frac{1}{n-3}}} \end{aligned} where \begin{aligned} U&=\frac{1}{2}\log_e \bigg(\frac{1+r}{1-r}\bigg) \end{aligned} and \begin{aligned} \xi & =\frac{1}{2}\log_e \bigg(\frac{1+\rho_0}{1-\rho_0}\bigg) \end{aligned}

The test statistic $Z$ follows standard normal distribution $N(0,1)$.

### Step 4 Determine the critical values

For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.

• For left-tailed alternative hypothesis: Find the $Z$-critical value using

\begin{aligned} P(Z<-Z_\alpha) &= \alpha. \end{aligned}

• For two-tailed alternative hypothesis: $Z_{\alpha/2}$.

$$P(Z<-Z_{\alpha/2} \text{ or } Z> Z_{\alpha/2}) = \alpha.$$

• For right-tailed alternative hypothesis: $Z_\alpha$. \begin{aligned} P(Z>Z_\alpha) & = \alpha. \end{aligned}

### Step 5 Computation

Compute the test statistic under the null hypothesis $H_0$ using equation \begin{aligned} Z_{obs} &= \dfrac{U-\xi}{\sqrt{\frac{1}{n-3}}} \end{aligned}

### Step 6 Decision (Traditional Approach)

Based on the critical values.

• For left-tailed alternative hypothesis: Reject $H_0$ if $Z_{obs}\leq -Z_\alpha$.
• For right-tailed alternative hypothesis: Reject $H_0$ if $Z_{obs}\geq Z_\alpha$.
• For two-tailed alternative hypothesis: Reject $H_0$ if $|Z_{obs}|\geq Z_{\alpha/2}$.

OR

### Step 6 Decision ($p$-value Approach)

It is based on the $p$-value.

Alternative Hypothesis Type of Hypothesis $p$-value
$H_a: \rho<\rho_0$ Left-tailed $p$-value $= P(Z\leq Z_{obs})$
$H_a: \rho>\rho_0$ Right-tailed $p$-value $= P(Z\geq Z_{obs})$
$H_a: \rho\neq \rho_0$ Two-tailed $p$-value $= 2P(Z\geq abs(Z_{obs}))$

If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.