Question 8.13: The article “Withdrawal Strength of Threaded Nails” (D. Ramm......
The article “Withdrawal Strength of Threaded Nails” (D. Rammer, S. Winistorfer, and D. Bender, Journal of Structural Engineering, 2001:442–449) describes an experiment to investigate the relationship between the diameter of a nail (x) and its ultimate withdrawal strength (y). Annularly threaded nails were driven into Douglas fir lumber, and then their withdrawal strengths were measured in N/mm. The following results for 10 different diameters (in mm) were obtained.
The sample correlation coefficient is computed to be r = 0.7492. Test the hypothesis H_{0} : \rho ≤ 0 versus H_{1} : \rho > 0.
| x | 2.52 | 2.87 | 3.05 | 3.43 | 3.68 | 3.76 | 3.76 | 4.50 | 4.50 | 5.26 |
| y | 54.74 | 59.01 | 72.92 | 50.85 | 54.99 | 60.56 | 69.08 | 77.03 | 69.97 | 90.70 |
Under H_{0} we take 𝜌 = 0, so the test statistic U has a Student’s t distribution with n−2 = 8 degrees of freedom. Since the sample correlation is r = 0.7492, the value of U is
U = \frac{r\sqrt{n\ −\ 2} }{\sqrt{1\ −\ r^{2}}}
= \frac{0.7492\sqrt{10\ −\ 2} }{\sqrt{1\ −\ 0.7492^{2}}}
= 3.199
Consulting the Student’s t table with eight degrees of freedom, we find that the P-value is between 0.005 and 0.01. (Software yields P = 0.0063.) It is reasonable to conclude that 𝜌 > 0.