In hypothesis testing for a population correlation coefficient, row, we use a sample of ordered pairs, like these, to calculate a standardized test statistic and reject or fail to
reject a null hypothesis depending on whether that test statistic falls in a rejection region. And this test tells us whether the correlation between 2 variables is significant.
The usual pairing of hypotheses are listed here, with the null hypothesis representing no significant correlation, and as with other hypothesis tests, they can be 1 or 2 tailed
tests. When testing for a population correlation coefficient, we use a t test, and the formula for calculating the standardized test statistic is t equals, r divided by,
the square root of 1 minus r squared over n minus 2. In this formula t is the standardized test statistic, r is the sample correlation coefficient,
which is calculated using this formula from a data set of ordered pairs, and n is the number of ordered pairs, and the degrees of freedom is equal n minus 2.
Alright, let’s go through a full example step by step. Here is a data set for video views on my YouTube channel and subscribers gained, for 16 randomly selected days
over the past year. With views being the x or independent variable and subscribers gained being the y or dependent variable. The r value, or sample correlation coefficient
equals 0.9067. At a level of significance of alpha equals 0.05, test the significance of this correlation coefficient. Step 1 is to identify the null and alternative hypotheses.
The null hypothesis, h sub 0 is row equals 0, that there is no significant correlation, and the alternative hypothesis is then row does not equal 0. Step 2 is to identify the
level of significance, which was given, alpha equals 0.05. Step 3 is to identify the degrees of freedom, which is n minus 2, so 14 degrees of freedom.
Step 4, is to determine the test to use, left tailed, right tailed or 2 tailed, and because the alternative hypothesis contains the does not equals inequality,
this will be a 2 tailed test. Step 5 is to determine the critical values. Graphically this looks like this, with our critical values, negative t naught and positive t naught,
here, and both of these shaded areas in the tails being rejection regions. Since the level of significance is alpha equals 0.05, and this is a 2 tailed test,
each of these rejection regions will be one half of alpha, which is 0.025. And in the t distribution table that value is -2.145 and positive 2.145, so those are our critical
values. Step 6 is to identify the rejection regions, and our rejection regions is any standardized test statistic value that falls in the shaded area, that is any value
that is less -2.145 or any value greater than positive 2.145. Step 7 is to use the formula and calculate the t value, or the value of the standardized test statistic.
And in our example, we have t equals 0.9067, divided by the square root of ,1 minus 0.9067 squared, over 16 minus 2. And this calculates out to 8.0452.
Step 8 is to make a decision to reject or fail to reject the null hypothesis. On our graph, you can see that the standardized test statistic does fall in the rejection region
to the right, as t, the standardized test statistic is greater than positive t naught, the critical value. So, in this case we would reject the null hypothesis.
Step 9 is to interpret the decision. There is enough evidence at the 5% level of significance to conclude that there is a significant linear correlation
between video views and subscribers gained. Alright my friends, that be the basics on hypothesis testing for a population correlation coefficient.
Hopefully you got something out of this video, I do have more statistical videos right there for you, till next time, I am outta here.