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

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

Explanation:

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

`\bar d=(1)/(n)\sum d_(i)=(1)/(15)* [1380+3370+2580+...+3520]=2642.67\\\\S_(d)=\sqrt{(1)/(n-1)\sum (d_(i)-\bar d)^(2)}\\=\sqrt{(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 (S_(d))/(√(n))`

`=2642.67+2.145\cdot (525.69)/(√(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.