Question

Can a correlation coefficient be statistically significant? If so, what would be the null hypothesis be? Please provide good, clear detail as if you were trying to explain the information in detail but enough for a 12 year old to understand

Answer 1

The t-test is used to establish if the correlation coefficient is significantly different from zero, and, hence that there is evidence of an association between the two variables. There is then the underlying assumption that the data is from a normal distribution sampled randomly.

Hypothesis :

• Null Hypothesis: H0: ρ = 0 The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population.
• Alternate Hypothesis: Ha: ρ ≠ 0 The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

Test Statistics :

The formula for the test statistic is,

t= r*√(n−2) / √(1−r^2)

It follows the distribution with n-2 degree of freedom

There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the p-value
• Method 2: Using a table of critical values

alpha = level of singnificance, simply it is an acceptable errror. We choose alpha 1%, 5%, 10% based on our test.

If p-value < alpha

Ho is rejected.

Therefore we conclude that correlation coefficients are significant.

Or

If test statistics > crtitical value

Ho is rejected.

Therefore we conclude that the correlation coefficients are significant.

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