Final answer:
To determine if the correlation in the population is greater than zero, we need to establish null and alternative hypotheses and perform a hypothesis test using the sample correlation coefficient and critical value. If the sample correlation coefficient is greater than the critical value, we reject the null hypothesis and conclude that there is a significant linear relationship between the variables in the population.
Step-by-step explanation:
When conducting a hypothesis test to determine if the correlation in the population is greater than zero, we need to establish null and alternative hypotheses. In this case, the null hypothesis (H0) is that the population correlation coefficient rho (p) is less than or equal to zero, while the alternative hypothesis (H1) is that the population correlation coefficient is greater than zero.
To test these hypotheses, we calculate the sample correlation coefficient r, which in this case is 0.32. We then compare this value to the critical value at a significance level of 0.05. If the sample correlation coefficient is greater than the critical value, we reject the null hypothesis and conclude that there is a significant linear relationship between the two variables in the population. However, if the sample correlation coefficient is less than or equal to the critical value, we fail to reject the null hypothesis and do not have sufficient evidence to conclude that the correlation in the population is greater than zero.
The student's question is about determining if the population correlation coefficient is greater than zero based on a sample correlation of .32 using hypothesis testing and a .05 significance level. To conclude, we compare the sample correlation with critical values or perform a t-test, and if results are significant, we reject the null hypothesis.
The student has asked whether we can conclude that the correlation in the population is greater than zero, given a random sample of 12 paired observations indicated a correlation of .32, and using a .05 significance level. To answer this question, we can use hypothesis testing for the significance of the correlation coefficient.
Hypothesis testing involves comparing the sample correlation coefficient (r) with a critical value from a table or using a t-test to determine if the correlation is significantly different from zero. For this example, the null hypothesis (H0) is that the population correlation coefficient (rho) is less than or equal to zero, and the alternative hypothesis (H1) is that it is greater than zero.
To test the null hypothesis, we would typically use a t-test for the correlation coefficient. However, since the critical value for the sample size is not provided, we would either look up the critical value for a correlation coefficient in a table for the given degrees of freedom (df = n - 2), or perform a linear regression t-test.
If the calculated t-value is greater than the critical t-value for the given degrees of freedom at the .05 significance level, or if the p-value is less than .05, we reject the null hypothesis and conclude there is a significant linear relationship.