Hypothesis Test for Correlation

Let's look at the hypothesis test for correlation, including the hypothesis test for correlation coefficient, the hypothesis test for negative correlation and the null hypothesis for correlation test.

What is the hypothesis test for correlation coefficient?

When given a sample of bivariate data (data which include two variables), it is possible to calculate how linearly correlated the data are, using a correlation coefficient.

The product moment correlation coefficient (PMCC) describes the extent to which one variable correlates with another. In other words, the strength of the correlation between two variables. The PMCC for a sample of data is denoted by r, while the PMCC for a population is denoted by ρ.

The PMCC is limited to values between -1 and 1 (included).

• If$r=1$, there is a perfect positive linear correlation. All points lie on a straight line with a positive gradient, and the higher one of the variables is, the higher the other.

• If$r=0$, there is no linear correlation between the variables.

• If$r=-1,$ there is a perfect negative linear correlation. All points lie on a straight line with a negative gradient, and the higher one of the variables is, the lower the other.

Correlation is not equivalent to causation, but a PMCC close to 1 or -1 can indicate that there is a higher likelihood that two variables are related.

Bivariate data with no correlation, positive correlation, and negative correlation

The PMCC should be able to be calculated using a graphics calculator by finding the regression line of y on x, and hence finding r (this value is automatically calculated by the calculator), or by using the formula$r=\frac{S_{xy}}{\sqrt{S_{xx}{S}_{yy}}}$, which is in the formula booklet. The closer r is to 1 or -1, the stronger the correlation between the variables, and hence the more closely associated the variables are. You need to be able to carry out hypothesis tests on a sample of bivariate data to determine if we can establish a linear relationship for an entire population. By calculating the PMCC, and comparing it to a critical value, it is possible to determine the likelihood of a linear relationship existing.

What is the hypothesis test for negative correlation?

To conduct a hypothesis test, a number of keywords must be understood:

• Null hypothesis ( $H_0$): the hypothesis assumed to be correct until proven otherwise

• Alternative hypothesis ( $H_1$): the conclusion made if$H_0$ is rejected.

• Hypothesis test: a mathematical procedure to examine a value of a population parameter proposed by the null hypothesis compared to the alternative hypothesis.

• Test statistic: is calculated from the sample and tested in cumulative probability tables or with the normal distribution as the last part of the significance test.

• Critical region: the range of values that lead to the rejection of the null hypothesis.

• Significance level: the actual significance level is the probability of rejecting$H_0$ when it is in fact true.

The null hypothesis is also known as the 'working hypothesis'. It is what we assume to be true for the purpose of the test, or until proven otherwise.

The alternative hypothesis is what is concluded if the null hypothesis is rejected. It also determines whether the test is one-tailed or two-tailed.

A one-tailed test allows for the possibility of an effect in one direction, while two-tailed tests allow for the possibility of an effect in two directions, in other words, both in the positive and the negative directions. Method: A series of steps must be followed to determine the existence of a linear relationship between 2 variables. 1. Write down the null and alternative hypotheses ($H_{0}and{H}_1$). The null hypothesis is always$\rho =0$, while the alternative hypothesis depends on what is asked in the question. Both hypotheses must be stated in symbols only (not in words).

2. Using a calculator, work out the value of the PMCC of the sample data, r .

3. Use the significance level and sample size to figure out the critical value. This can be found in the PMCC table in the formula booklet.

4. Take the absolute value of the PMCC and r, and compare these to the critical value. If the absolute value is greater than the critical value, the null hypothesis should be rejected. Otherwise, the null hypothesis should be accepted.

5. Write a full conclusion in the context of the question. The conclusion should be stated in full: both in statistical language and in words reflecting the context of the question. A negative correlation signifies that the alternative hypothesis is rejected: the lack of one variable correlates with a stronger presence of the other variable, whereas, when there is a positive correlation, the presence of one variable correlates with the presence of the other.

How to interpret results based on the null hypothesis

From the observed results (test statistic), a decision must be made, determining whether to reject the null hypothesis or not.

Image: Repapetilto CC BY-SA 3.0,

Two-tailed test applied to normal distribution. Image: public domain

Both the one-tailed and two-tailed tests are shown at the 5% level of significance. However, the 5% is distributed in both the positive and negative side in the two-tailed test, and solely on the positive side in the one-tailed test.

From the null hypothesis, the result could lie anywhere on the graph. If the observed result lies in the shaded area, the test statistic is significant at 5%, in other words, we reject$H_0$. Therefore,$H_0$ could actually be true but it is still rejected. Hence, the significance level, 5%, is the probability that$H_0$ is rejected even though it is true, in other words, the probability that$H_0$ is incorrectly rejected. When $H_0$ is rejected, $H_1$(the alternative hypothesis) is used to write the conclusion.

We can define the null and alternative hypotheses for one-tailed and two-tailed tests:

For a one-tailed test:

• $H_0:\rho =0:H_1\rho >0or$
• $H_0:\rho =0:H_1\rho <0$

For a two-tailed test:

• $H_0:\rho =0:H_1\rho \ne 0$

Let us look at an example of testing for correlation.

Let us look at a second example.

ImagesOne-tailed test: https://en.wikipedia.org/w/index.php?curid=35569621