Inference for Regression A fisheries biologist has been stud

Inference for Regression A fisheries biologist has been studying horseshoe crabs. She has sampled 100 horseshoe crabs and recorded their weight (in kilograms) and width (in centimeters). The proposed regression equation is

weight = b + width * m

This model was fit to the data using the method of least squares. The following results were obtained from statistical software.

Variable

Estimate

Std. error of estimate

Intercept (b)

2.3013

0.9788

Slope (m)

0.7963

0.0939

R2 = 0.423

A.) What is the regression equation for this example?

B.) What is the explanatory, or predictor, variable in this study?

C.) If the researcher wanted to test whether there is a statistically significant relationship between these two variables, what would the test statistic be? Calculate it from the table above.

D.) What can we say about the p-value?

E.) Ultimately, the reason that we find test statistics is so that we can compare them to a null distribution.For regression, that is a t-distribution based on the degrees of freedom.With 98 degrees of freedom (100-2), we can safely say that the critical t (or the confidence multiplier) is what?

F.) Find the confidence interval for the slope.

G.) Is there a statistically significant relationship?Answer with the test statistic and the confidence interval.

Solution

In case of multiple parts in the question, the first 4 parts will be answered.

A. We use the regression estimates to determine the equation:

Y = b + m*X

b = 2.3013

m = 0.7963

So , regression equation:

Y = 2.3013 + 0.7963*X

B. Predictor variable (X) : Width of horseshoe crabs

C. We test is m = 0

Test statistic: Estimate of m / Std. error of m

= 0.7963 / 0.0939 = 8.4803

D. P - Value: 2* P(t_100-2 > 8.4803) = nearly zero

P - value is very small, so we reject the null hypothesis.

So b is significantly different from zero.