The data given provides shear strengths for I-beams. Column A is from the I-beam

Question

The data given provides shear strengths for I-beams. Column A is from the I-beam currently used, column B is from a proposed substitute. Prove if the shear strength for the proposed substitute is greater than that currently used (90% confidence level).

   A                                                       B

2110.46 2003.898 2083.236 2161.299 1894.69 2070.339 1899.14 2078.5 1979.78 2050.213 2020.605 2236.334 1919.594 1987.827 1939.767 1895.548 2008.881 2098.094 1965.564 2024.152 2039.726 2024.725 2007.004 2048.784 1899.521 2069.437 2090.526 1916.153 2008.903 2048.208 1709.704 1984.075 1989.777 2074.655 1784.298 1899.8

Explanation / Answer

Set Up Hypothesis
Null, shear strength for the proposed substitute is similar Ho: u1 = u2
Alternate, shear strength for the proposed substitute is greater than current - H1: u1 < u2
Test Statistic
X(Mean)=1969.78
Standard Deviation(s.d1)=101.66 ; Number(n1)=20
Y(Mean)=2043.59
Standard Deviation(s.d2)=83.181; Number(n2)=16
we use Test Statistic (t) = (X-Y)/Sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =1969.78-2043.59/Sqrt((10334.7556/20)+(6919.07876/16))
to =-2.4
| to | =2.4
Critical Value
The Value of |t | with Min (n1-1, n2-1) i.e 15 d.f is 1.753
We got |to| = 2.39575 & | t | = 1.753
Make Decision
Hence Value of | to | > | t | and Here we Reject Ho
P-Value:Left Tail - Ha : ( P < -2.3957 ) = 0.01504
Hence Value of P0.05 > 0.01504,Here we Reject Ho

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