Question

A study of 11 woridwide financial institutions showed the correlation between their assets and pretax profit to be 0.94. At the 0.01 significance level, can we conclude that there is positive correlation in the population? (Round the final answer to 3 decimal places.) H0:p≤0 H:rho>0 Reject H 0 ​ if t>2.821 t= , there is a correlation between assets and pretax proft.

Answer 1

Final answer:

At the 0.01 significance level, with a calculated t-statistic of 8.268 (greater than 2.821), there is sufficient evidence to conclude a positive correlation between assets and pretax profit in the population.

Explanation:

To test whether there is a positive correlation in the population, we can use a one-tailed hypothesis test for correlation. The null hypothesis
`(\(H_0\))` is that there is no correlation
`(\(\rho \leq 0\))`, and the alternative hypothesis
`(\(H\))` is that there is a positive correlation
`(\(\rho > 0\))`.

Given that the sample correlation coefficient
`(\(r\))` is 0.94 and the sample size
`(\(n\))` is 11, we can use the formula for the t-statistic for correlation:

`\[ t = (r √(n-2))/(√(1-r^2)) \]`

Substitute the values into the formula:

`\[ t = (0.94 √(11-2))/(√(1-0.94^2)) \]`

Now, calculate the t-statistic:

`\[ t \approx (0.94 √(9))/(√(1-0.8836)) \]`

`\[ t \approx (0.94 * 3)/(√(0.1164)) \]`

`\[ t \approx (2.82)/(0.341) \]`

`\[ t \approx 8.268 \]`

Now, compare the calculated t-statistic (8.268) with the critical t-value for a one-tailed test with 9 degrees of freedom (11 - 2 = 9) and a significance level of 0.01.

If
`\(t > 2.821\)`, we reject the null hypothesis. In this case,
`\(8.268 > 2.821\)`, so we reject
`\(H_0\)`. Therefore, there is sufficient evidence at the 0.01 significance level to conclude that there is a positive correlation between assets and pretax profit in the population.