Question

A study of 20 worldwide financial institutions showed the correlation between their assets and pretax profit to be 0.81.State the decision rule for 0.025 significance level: H0: rho ≤ 0; H1: rho > 0. (Round your answer to 3 decimal places.)Compute the value of the test statistic. (Round your answer to 2 decimal places.)Can we conclude that the correlation in the population is greater than zero? Use the 0.025 significance level.

Answer 1

Answer:

t=5.86

We can conclude that the population correlation between their assets and pretax profit is higher than 0 at the significance level provided.

Step-by-step explanation:

n= 20 random sample taken

r=0.81 correlation coeffcient obtained

$\alpha=0.025$ significance level obtained

1) System of hypothesis

The system of hypothesis given are:

Null hypothesis :$\rho \leq 0$

Alternative hypothesis: $\rho >0$

2) Calculate the statistic

The statistic in order to test an hypothesis for the correlation coefficient is given by:

$t =\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

This statistic follows a t distribution with n-2 degrees of freedom

If we replace the values given we got:

$t =\frac{0.81\sqrt{20-2}}{\sqrt{1-(0.81^2)}}=5.86$

3) P value

For this case w eneed to calculate first the degrees of freedom

$df=n-2=20-2=18$

And then analyzing the alternative hypothesis we can calculate the p value on this way:

$p_v =P(t_{18} >5.86) =1-P(t_{18} <5.86)=1-0.99999=7.51x10^{-6}$

Since the P-value is smaller than the significance level, we have enough evidence to reject the null hypothesis in favor of the alternative. We conclude "there is sufficient evidence at the significance level to conclude that there is a linear relationship in the population between the two variables analyzed."