Final answer:
After carrying out the t test using the provided regression output, the result shows that with a p-value of 0.026, which is lower than the alpha level of 0.05, we reject the null hypothesis, concluding that there is a significant linear relationship between x and y.
Step-by-step explanation:
The question asks us to determine whether or not there is a significant linear relationship between the independent variable x and the dependent variable y using a t test, given a significance level (alpha) of 0.05.
Performing the t test
We begin by stating our null hypothesis H0: there is no linear relationship between x and y (the slope of the regression line is equal to zero). The alternative hypothesis Ha: there is a significant linear relationship (the slope is not equal to zero).
From the provided computer output, we have a t Stat of -0.903 for the slope coefficient of the variable x. The standard error associated with this coefficient is 0.26.
To perform the t test, we compare the absolute value of the t Stat to the critical value from the t-distribution table for the given degrees of freedom, which is 13 (df = n - 2, since n = 15 which is the number of observations). For an alpha level of 0.05 and 13 degrees of freedom, the two-tailed test critical value is approximately 2.160. Since the absolute value of our t Stat (-0.903) is not greater than the critical value, this suggests that the slope coefficient is not significantly different from zero.
However, we also have the p-value, which is 0.026. The p-value tells us the probability of observing a test statistic as extreme as the t Stat, under the assumption that the null hypothesis is true. Since the p-value is less than alpha, we reject the null hypothesis.
Conclusion
There is sufficient evidence to conclude that there is a significant linear relationship between x and y. Therefore, we can say that y and x are related.