Question

In 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), the following measurements were given for the cylindrical compressive strength (in MPa) for 11 beams:

38.43

38.43

38.39

38.83

38.45

38.35

38.43

38.31

38.32

38.48

38.5

One thousand bootstrap samples were generated from these data, and the bootstrap sample means were arranged in order. Refer to the smallest value as Y₁, the second smallest as Y₂, and so on, with the largest being Y₁₀₀₀. Assume that Y₂₅ = 38.3818, Y₂₆ = 38.3818, Y₅₀ = 38.3909, Y₅₁ = 38.3918, Y₉₅₀ = 38.5218, Y₉₅₁ = 38.5236, Y₉₇₅ = 38.5382, and Y₉₇₆ = 38.5391.

Compute a 90% bootstrap confidence interval for the mean compressive strength, using the expression (2X⎯⎯⎯−X⎯⎯⎯∗1−α/2, 2X⎯⎯⎯−X⎯⎯⎯∗α/2)(2X¯−X¯1−α/2∗, 2X¯−X¯α/2∗). Round the answers to four decimal places.

The 90% bootstrap confidence interval for the mean compressive strength is ( , ).

Answer 1

The mean of the given sample is 38.4473.

Let us compute $\delta^*=\bar{x}^*-\bar{x}$ for each sample which is given in the below table

sample mean	$\delta^*$
. .
y25=38.3818	38.3818-38.4473=-0.0655
Y26	-0.0655
. .
Y50	-0.0564
Y51	-0.0555
. .
Y950	0.0745
Y951	0.0763
. .
Y975	0.0909
Y976	0.0918

Given
α = 0.10
then
a/2 = 0.05
and
1000 a/2 50 *
; and
1000 * (1-a/2) 950
we pick the $\delta^*$ values for Y50 to calculate the upper limit and Y950 for calculating the lower limit.

Please Note: The expression given above was not clear to me. Please comment incase if you have doubts

The 95% bootstrap confidence interval is given by (38.3728,38.5037)

0.950, x-