Question

As part of a larger project to study the behavior of stressed-skin panels, a structural component being used extensively in North America, an article reported on various mechanical properties of Scotch pine lumber specimens. Data on the modulus of elasticity (MPa) obtained 1 minute after loading in a certain configuration and 4 weeks after loading for the same lumber specimens is presented here.
Observatio 1 min 4 Week Difference
1 16,620 9,110 1380
2 17,300 13,250 3370
3 15,480 14,720 2580
4 12,970 12,740 2740
5 17,260 10,120 2850
6 13,400 14,570 2690
7 13,900 11,220 2180
8 13,630 11,100 2800
9 13,260 11,420 2210
10 14,370 10,910 2350
11 11,700 12,110 2260
12 15,470 8,620 3080
13 17,840 12,590 2880
14 14,070 15,090 2750
15 14,760 10,550 3520

Required:
Calculate and interpret an upper confidence bound for the true average difference between 1-minute modulus and 4-week modulus; first check the plausibility of any necessary assumptions. (Use α = 0.05. Round your answer to the nearest whole number.)

Answer 1

Answer:

The upper confidence bound for the true average difference between 1-minute modulus and 4-week modulus is 2933.82.

Step-by-step explanation:

Compute the mean difference and standard deviation of the difference as follows:

$\bar d=\frac{1}{n}\sum d_{i}=\frac{1}{15}\times [1380+3370+2580+...+3520]=2642.67\\\\S_{d}=\sqrt{\frac{1}{n-1}\sum (d_{i}-\bar d)^{2}}\\=\sqrt{\frac{1}{15-1}[(1380-2642.67)^{2}+(3370-2642.67)^{2}+...}=525.69$

The degrees of freedom is:

df = n - 1

   = 15 - 1

   = 14

Th critical value of t is:

$t_{\alpha/2, (n-1)}=t_{0.05/2, 14}=2.145$

*Use a t-table.

Compute the upper confidence bound for the true average difference between 1-minute modulus and 4-week modulus as follows:

$\text{Upper Confidence Bound}=\bar d+t_{\alpha/2, (n-1)}\cdot \frac{S_{d}}{\sqrt{n}}$

                                        $=2642.67+2.145\cdot \frac{525.69}{\sqrt{15}}\\\\=2642.67+291.15\\\\=2933.82$

Thus, the upper confidence bound for the true average difference between 1-minute modulus and 4-week modulus is 2933.82.