Question 5.18: The article “Direct Strut-and-Tie Model for Prestressed Deep......
The article “Direct Strut-and-Tie Model for Prestressed Deep Beams” (K. Tan, K. Tong, and C. Tang, Journal of Structural Engineering, 2001:1076–1084) presents measurements of the nominal shear strength (in kN) for a sample of 15 prestressed concrete beams. The results are
Is it appropriate to use the Student’s t distribution to construct a 99% confidence in-terval for the mean shear strength? If so, construct the confidence interval. If not, explain why not.
| 550 | 920 | 875 | 850 | 825 | 428 | 400 | 580 |
| 950 | 735 | 590 | 360 | 636 | 750 | 575 | |
| Source: Journal of Structural Engineering. | |||||||
To determine whether the Student’s t distribution is appropriate, we will make a boxplot and a dotplot of the sample. These are shown in the following figure.
There is no evidence of a major departure from normality; in particular, the plots are not strongly asymmetric, and there are no outliers. The Student’s t method is appropriate. We therefore compute \overline{X} = 668.27 and s = 192.089. We use expression (5.14) with n = 15 and 𝛼 ∕ 2 = 0.005. From the t table with 14 degrees of freedom, we find t_{14,.005} = 2.977. The 99% confidence interval is 668.27 ± (2.977)(192.089) ∕ \sqrt{15}, or (520.62, 815.92).
Let X_{1},…, X_{n} be a small random sample from a normal population with mean 𝜇. Then a level 100(1 − 𝛼)% confidence interval for 𝜇 is
\overline{X} ± t_{n-1 , α/2} \frac{s}{\sqrt{n}} (5.14)
