Question

A sample of 6 head widths of seals (in cm) and the corresponding weights of the seals (in kg) were recorded. Given a linear correlation coefficient of 0.948, find the corresponding critical values, assuming a 0.01 significance level. Is there sufficient evidence to conclude that there is a linear correlation?

Answer 1

Answer:

We conclude that there is not a linear correlation.

Step-by-step explanation:

We are given that a sample of 6 head widths of seals (in cm) and the corresponding weights of the seals (in kg) was recorded.

Given a linear correlation coefficient of 0.948.

Let ρ = population correlation coefficient.

So, Null Hypothesis, $H_0$ : ρ = 0     {means that there is a linear correlation}

Alternate Hypothesis, $H_A$ : ρ $\neq$ 0      {means that there is not a linear correlation}

The test statistics that would be used here One-sample t-test statistics for testing population correlation coefficient;

                            T.S. =  $\frac{r \times \sqrt{n-2} }{\sqrt{1 -r^{2} } }$  ~  $t_n_-_2$

where, r = sample correlation coefficient = 0.948

           n = sample of head widths of seals = 6

So, the test statistics  =  $\frac{0.948 \times \sqrt{6-2} }{\sqrt{1 -0.948^{2} } }$  ~ $t_4$ 

                                     =  5.957

The value of t-test statistics is 5.957.

Now at 0.01 level of significance, the t table gives a critical value of 4.604 at 4 degrees of freedom for a two-tailed test.

Since our test statistics is more than the critical value of t as 5.957 > 4.604, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that there is not a linear correlation.