Question

If there had been$\mathbf{n}\mathbf{=}\mathbf{13}$replications in both data sets, would the difference in standard deviations be significant?

Short answer

Here,${\mathrm{F}}_{\mathrm{calculated}}=2.82>{\mathrm{F}}_{\mathrm{table}}=2.69.$.

Hence, the differences in the standard deviation of the two given sets are significant.

Step-by-step solution

F test

F-test:${\mathrm{F}}_{\mathrm{calculated}}=\frac{{{\mathrm{S}}_1}^2}{{{\mathrm{Z}}_2}^2}$

Using F-test

The number of replications in both data sets,

Using Table 4-2, we know that,

F-Test:

On considering larger a larger standard deviation in the numerator, we. We have:,

$$\begin{gathered} s_1=0.47 \\ {s}_1=0.28 \end{gathered}$$

Using F-test, the$\mathrm{F}$ is calculated as follows:,

role="math" localid="1663326924581" $$\begin{gathered} {\mathrm{F}}_{\mathrm{calculated}}=\frac{{{\mathrm{s}}_1}^2}{{{\mathrm{s}}^2}_1} \\ =\frac{{\left(0.47\right)}^2}{{\left(0.28\right)}^2} \\ =2.82. \end{gathered}$$

Degree of Freedom for Both

The degrees of freedom for ${\mathrm{s}}_1$are$\left(\mathrm{n}-1\Rightarrow 13-1=12\right).$

From Table 4-3, by locating, the column with 12 degrees of freedom for role="math" localid="1663326886623" $s_1$and the row with 12 degrees of freedom for $s_2$give.

Compare ${\mathrm{F}}_{\mathrm{calculated}}$with ${\mathrm{F}}_{\mathrm{table}}$.

${\mathrm{F}}_{\mathrm{calculated}}=2.82>{\mathrm{F}}_{\mathrm{table}}=2.69.$

Therefore, the differences in the standard deviation of the two given sets are significant.