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Chapter 9: Problem 12

Test the correlation, as indicated. Show all details of the test. Test for evidence of a linear association; \(r=0.28 ; n=10\)

Short Answer

Expert verified
We fail to reject the null hypothesis, confirming that there is not enough statistical evidence to suggest that there exists a linear association.

Step by step solution

01

Understand the Provided Details

Analyzing the data provided, it can be seen that the correlation coefficient \(r\) is stated as 0.28 and the sample size \(n\) is 10. The correlation coefficient lies within the permissible range of -1 to 1 inclusive.
02

Test for Linear Association

To test the hypothesis, let's consider a two-tail test. The null hypothesis \(H_0\) will be that the population correlation coefficient is zero (\(蟻 = 0\)), meaning there is no linear association. The alternative hypothesis \(H_1\) will be that the population correlation coefficient isn't zero (\(蟻 鈮� 0\)), indicating evidence of a linear association.
03

Compute Test Statistic

The formula to calculate test statistic is \(t = \frac{r \sqrt{n-2}}{\sqrt{1 - r^2}} \). Substituting the given values, we get \(t = \frac{0.28 \sqrt{10-2}}{\sqrt{1 - 0.28^2}} = 0.797\). This is the observed value of the test statistic.
04

Examine the Critical Value

With a degree of freedom of \(n-2 = 10-2 = 8\) and an alpha level of 0.05 for a two-tailed test, refer to an appropriate statistical table (such as the Student's T-distribution table) to find the critical value. The tabulated (critical) values for the t-test for a two-tailed test with 8 degrees of freedom at a 0.05 significance level are -2.306 and 2.306. If the computed statistic falls within this critical region, we reject the null hypothesis.
05

Make Decision on the Hypothesis

The observed test statistic 0.797 falls within the critical region of -2.306 and 2.306. Thus, we fail to reject the null hypothesis. This suggests that there is not enough evidence to support the claim that there is a linear association.

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