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 meas

Measurement
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?

2021-02-06
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:
$$n_{1}=21, s_{1}=3.06\ and\ n_{2}=21, s_{2}=1.41$$
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:
$$H_{0}:\sum_{1}^{2}=\sigma_{2}^{2}$$
Against the alternative hypotesis as
$$H_{0}:\sum_{1}^{2}\neq \sigma_{2}^{2}$$
This hypothesis follows a F statistics with $$df_{1}=n_{1}-1=20,df_{2}=n_{2}-1=20$$
and the confidence level at $$\alpha = 0.05$$:
Step 2
$$F=\frac{S_{1}^{2}}{S_{2}^{2}}=\frac{9.832}{2.088}=4.709$$
Where
$$S_{1}^{2}=\frac{n_{1}}{n_{1}-1}s_{1}^{2}=9.832\ and\ S_{2}^{2}=\frac{n_{2}}{n_{2}-1}s_{2}^{2}=2.088$$
Tabulated F statistics value is determined as:
$$F_{20,20,0.05}=2.124$$
As the calculated value is more that the tabulated value
$$F_{observed}=4.709>F_{20,20,0.05}=2.124$$
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.