## 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 3 Specify the level of significance $\alpha$

### 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.