Chapter 12, Problem 1CR

(a) How does correlation analysis differ from regression analysis? (b) What does a correlation coefficient reveal? (c) State the quick rule for a significant correlation and explain its limitations. (d) What sums are needed to calculate a correlation coefficient? (e) What are the two ways of testing a correlation coefficient for significance?

To determine

Explain the difference between correlation analysis and the regression analysis.

Explanation of Solution

Correlation Coefficient:

The correlation coefficient, r, between ordered pairs of variables, (x, y) having sample standard deviations $s_x$, $s_y$ and sample covariance $s_{xy}$ for a sample of size n is given as $r=\frac{s_{xy}}{s_x{s}_y}$. The correlation coefficient is useful for measuring the strength of the linear relationship between the two variables.

Regression:

Suppose $x_1\mathrm{...}{x}_n$ be n sample values of independent variable and the corresponding dependent variable values are $y_1\mathrm{...}{y}_n$. The slope and the intercept of ordinary least square can be defined as $b_0=\overline{y}-b_1\overline{x}$ and $b_1=\frac{S{S}_{xy}}{S{S}_{xx}}$.

Where, $S{S}_{xx},S{S}_{yy},S_{xy}$ are the sum of squares due to x, y and xy respectively. $\overline{x}\text{and}\overline{y}$ are the sample mean of the independent and dependent variable respectively.

From the regression the fitted line is denoted as, $\widehat{y}=b_0+b_{1}x$ .

Correlation analysis reveals the linear relationship between the independent and the dependent variables. But in regression analysis, the relation between the two variables can be anything not only linear. In this case, a value of dependent variable can be predicted for a particular value of independent variable.

To determine

Explain what actually a correlation coefficient reveals.

Explanation of Solution

The correlation coefficients reveals the linear relationship between the concern variables.

To determine

State the quick rule for a significant correlation.

Explain the limitations.

Explanation of Solution

Quick rule of correlation:

If the critical values of t is unavailable then the correlation coefficient (r) will be significant for,

$\left|r\right|>\frac{2}{\sqrt{n}}$ , where n is the sample size and the level of significance is 0.05.

Limitation of the quick rule:

For less sample of size, the quick rule of correlation can’t provide exact result.

If the sample size is more than 1,000 then the result will be acceptable.

To determine

Explain the sums are needed for finding the correlation.

Explanation of Solution

Formula for correlation coefficient:

Suppose $x_1\mathrm{...}{x}_n$ be n sample values of independent variable and the corresponding dependent variable values are $y_1\mathrm{...}{y}_n$. In addition, $\overline{x}\text{and}\overline{y}$ are the sample mean of the independent and dependent variables respectively. The sample correlation can be defined as,

$r=\frac{{\displaystyle \sum _{i=1}^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$

Therefore, to find the correlation coefficient, ${\displaystyle \sum _{i=1}^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}$,${\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}$ and ${\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}$ are needed.

To determine

Explain the two ways for testing a correlation coefficient for significance.

Explanation of Solution

There are two methods for concluding the significance. Let the level of significance is α and the sample size is n.

p-value method:

If $p\text{-value}\le \alpha$ then reject the null hypothesis.

If $p\text{-value >}\alpha$, then fail to reject the null hypothesis.

Critical value method:

Decision rule:

For two tailed test:

• If $t_{cal}>t_{\frac{\alpha }{2},\left(n-2\right)}\text{or }{t}_{cal}<-t_{\frac{\alpha }{2},\left(n-2\right)}$, reject the null hypothesis.
• Otherwise do not reject the null hypothesis.

For right tailed test:

• If $t_{cal}>t_{\alpha ,\left(n-2\right)}$, reject the null hypothesis.
• Otherwise do not reject the null hypothesis.

For left tailed test:

• If $t_{cal}<-t_{\alpha ,\left(n-2\right)}$, reject the null hypothesis.
• Otherwise do not reject the null hypothesis.

If the null hypothesis is rejected, the correlation coefficient is significant.