Question

Researchers studying pleasant touch sensations measured the firing frequency (impulses per second) of nerves that were stimulated by a light brushing stroke on the forearm and also recorded the subject's numerical rating of how pleasant the sensation was. The data from the study is below.

Firing Frequency	Pleasantness Rating
23	.2
24	1.0
22	1.2
25	1.2
27	1.0
28	2.0
34	2.3
33	2.2
36	2.4
34	2.8

Carry out a hypothesis test to decide if there is convincing evidence of a useful linear relationship between firing frequency and pleasantness rating. Use .05 as the level of significance.  

Be sure to state the hypotheses, name the test you are using, give the test statistic and p-value, and to make your conclusion.

Answer 1

Here we want to see that whether there is a linear relationship between Firing Frequency(xi's) and Pleasantness Rating(yi's)

So we will do correlation analysis to see that their is a linear relationship between the two or not.

Steps to do correlation analysis:

Step 1

where is the population correlation coefficent.

Step 2:

The sample statistic here will follow t distribution with n-2 df

where n is the number of observations.

Step 3:

Here the criteria for the decision is that if calculated |t| is more than tabulated t then we have enough evidence to reject the null hypothesis.

here it will be t distribution with n-2 = 10-2 = 8 df

so critical value of t distribution with 8 df and is given by:

tcritical, 0.05/2 = 2.306004 (Calculated in R)

So if |t| > tcritical, 0.05/2 = 2.306004  

Then we have enough evidence to reject H0

Step 4:

So the test statistic is given by:

where rxy is sample correlation.

and

and hence

t = 5.45180309

here
|t| > tcritical

And here

which implies that we do have enough evidence to reject H0

i.e., correlation between x and Y is not equal to zero.

which implies that their is a correlation between Firing Frequency and Pleasantness Rating