Before surgical removal of a diseased parathyroid gland, two tests are often performed: the standard intact test and the turbo test. Both tests measure parathyroid hormone (PTH, in ng/L) , but the turbo test is very expensive. Researchers obtained data from both tests in a sample of 48 patients to predict turbo test results (y) from standard intact test results (x) . The published findings include a scatterplot showing a clear linear relationship and the following summary:
Y = 1.08x - 4.36 (r = 0.97; n = 48)
We want to test a hypothesis of no relationship between the two procedures, H₀: p= 0 versus H_a: p≠0.
What is the P-value for this test?

Question

Quizplus · Accepted Answer

To test the hypothesis, we need to calculate the test statistic and find its corresponding P-value. Here, the null hypothesis is that there is no relationship between the two procedures, which means the slope of the regression line is zero (p=0). The alternative hypothesis is that there is a significant relationship between the two procedures (p≠0).

From the given information, the regression equation is: Y = 1.08x - 4.36
The slope of the regression line is 1.08, which is not zero. 
The correlation coefficient (r) is given as 0.97, which indicates a strong positive linear relationship between x and y.

To test the hypothesis, we can use a t-test for the slope of the regression line:
t = (b1 - 0) / SE(b1)
where b1 is the slope of the regression line and SE(b1) is the standard error of the slope.

From the regression output, we can calculate:
b1 = 1.08
SE(b1) = 0.054

Substituting these values, we get:
t = (1.08 - 0) / 0.054 = 20

The degrees of freedom for the t-test is n-2 = 48-2 = 46. 
Using a two-tailed test and a significance level of 0.05, the critical t-value is ± 2.013.

The calculated t-value of 20 is much larger than the critical t-value of ± 2.013, which means the test statistic falls in the rejection region. 
Therefore, we reject the null hypothesis and conclude that there is a significant relationship between the two procedures.

Since the P-value is less than 0.001, we choose A as the answer.

Before Surgical Removal of a Diseased Parathyroid Gland, Two Tests