Twenty-one independent measurements were taken of the hardness (on the Rockwell C scale) of HSLA-100 steel base metal, and another 21 independent measurements were made of the hardness of a weld produced on this base metal. The standard deviation of the measurements made on the base metal was 3.06, and the standard deviation of the measurements made on the weld was 1.41. Assume that the measurements are independent random samples from normal populations. Need to conclude that measurements made on the base metal are more variable than measurements made on the weld?

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Twenty-one independent measurements were taken of the hardness (on the Rockwell C scale) of HSLA-100 steel base metal, and another 21 independent measurements were made of the hardness of a weld produced on this base metal.   The standard deviation of the measurements made on the base metal was 3.06, and the standard deviation of the measurements made on the weld was 1.41.   Assume that the measurements are independent random samples from normal populations.   Need to conclude that measurements made on the base metal are more variable than measurements made on the weld?

Leonard Stokes · Accepted Answer

Step 1  According to the given question, twenty-one independent measurements were taken of the hardness (on the Rockwell C scale) of HSLA-100 steel base metal, and another 21 independent measurements were made of the hardness of a weld produced on this base metal.   The standard deviation of the measurements made on the base metal was 3.06, and the standard deviation of the measurements made on the weld was 1.41.   Assume that the measurements are independent random samples from normal populations.  Therefore the given data are as follows:  n1=21,s1=3.06 and n2=21,s2=1.41   In order to test whether the measurements made on the base metal are move variable than measurements made on the weld, we define the test hypothesis as:  H0:∑12=σ22  Against the alternative hypotesis as  H0:∑12≠σ22  This hypothesis follows a F statistics with df1=n1−1=20,df2=n2−1=20  and the confidence level at α=0.05:  Step 2  F=S12S22=9.8322.088=4.709  Where   S12=n1n1−1s12=9.832 and S22=n2n2−1s22=2.088  Tabulated F statistics value is determined as:  F20,20,0.05=2.124  As the calculated value is more that the tabulated value  Fobserved=4.709&amp;gt;F20,20,0.05=2.124  We reject the null hypothesis at 55 level of significance and we have  sufficient evidence to conclude that measurements made on the base metal are more variable than measurements made on the weld.

Twenty-one independent measurements were taken of the hardness (on the Rockwell C scale) of HSLA-100 steel base metal, and another 21 independent meas

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