Example: Calculate the correlation coefficient of:

Method

Use a statistical calculator

• A. Select "Statistics" mode.
• B. Choose the "$y = a + bx$" option for linear regression.
• C. Enter the data values.
• D. Select "Regression Calc".
• E. The correlation coefficient $r$ is calculated as $r = 0.845$.

Example: Test at the $2.5$% significance level whether the two populations are positively correlated. Step 1: Write out the null and alternate hypotheses:

• $H_0: \rho = 0$ (where $\rho$ represents the correlation coefficient of the population, whereas $r$ is the correlation coefficient of just the sample).
• $H_1: \rho > 0$ (the alternate hypothesis).
• $H_0$ is always "no correlation."

Step 2: Calculate $r$

Step 3: Check this value of $r$ against critical values in the table and conclude:

To be $97.5$% certain ($2.5$% significance/uncertainty), we must see $r > 0.8783$ to reject $H_0$

If testing for negative correlation, $r = -0.8783$.

If testing for general correlation, calculate $r$ and apply the above rules based on whatever sign $r$ has.

Since $r = 0.8450 < 0.8783$:

• Do not reject $H_0$.
• Conclusion: Insufficient evidence to suggest that the two populations are correlated positively.

Example: A computer-controlled milling machine is calibrated between $1$ and $7$ times a week. A supervisor recorded the number of weekly calibrations, $x$, and the number of manufacturing errors, $y$, in each of $7$ weeks.

(a) Calculate the product moment correlation coefficient for these data.

Use a statistical calculator:

• Select "Statistics" mode.
• Choose the "$y = a + bx$" option for linear regression.
• Enter the data values.
• Select "Regression Calc".
• The correlation coefficient $r$ is calculated as $r = 0.9599$.

(b) For these data, test $H_0: \rho = 0$ against $H_1: \rho \neq 0$, using a $1$% significance level.

Step 1: Check $r$ against the critical values in the table:

We need to compare r with the critical values at the $1$% significance level for a one-tailed test.

For a sample size of $n = 7$, the critical value from the table is:

Critical value = $0.8745$ (from the 1-tail, 1% column for $n = 7$)

Step 2: Conclude

Reject $H_0$.

There is sufficient evidence to suggest that the populations are correlated.