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Hypotheses are typically written in the form of if/then statements, such as if someone consumes a lot of sugar, they will develop cavities in their teeth.

Given:

\(I = 4\)

Let assume

\(\alpha = 0.05\)

The null hypothesis states that all population means are equal:

\({H_0}\,:\,\,{\mu _1} = {\mu _2} = {\mu _3} = {\mu _4}\)

The alternative hypothesis states the opposite of the null hypothesis

\({H_1};\)Not all of \({\mu _1},{\mu _2},{\mu _3},{\mu _4}\)are equal

The overall mean is the sum of all values divided by the number of values:

\(\overline x = \frac{{{n_1}{{\overline x }_1} + {n_2}{{\overline x }_2} + ...{n_k}{{\overline x }_k}}}{N}\)

\(\begin{aligned}{l} = \frac{{2\left( {29.8} \right) + 2\left( {27.4} \right) + 2\left( {25.95} \right) + 2\left( {27.15} \right)}}{{24}}\\ \approx 27.575\end{aligned}\)

The mean square for treatment is:

\(MS{T_r} = \frac{{{n_1}{{\left( {\overline {{x_1}} - \overline x } \right)}^2} + {n_2}{{\left( {\overline {{x_2}} - \overline x } \right)}^2} + ... + {n_k}{{\left( {\overline {{x_k}} - \overline x } \right)}^2}}}{{I - 1}}\)

\( = \frac{{2{{\left( {29.8 - 27.575} \right)}^2} + 2{{\left( {27.4 - 27.575} \right)}^2} + 2{{\left( {25.95 - 27.575} \right)}^2} + 2{{\left( {27.15 - 27.575} \right)}^2}}}{{4 - 1}}\)

\( = 5.2017\)

Mean square for error is:

\(MSE = \frac{{\left( {{n_1} - 1} \right){s_1}^2 + \left( {{n_2} - 1} \right){s_2}^2 + \left( {{n_k} - 1} \right){s_k}^2}}{{N - I}}\)

\( = \frac{{(2 - 1){{\left( {0.8485} \right)}^2} + (2 - 1){{\left( {0.4243} \right)}^2} + (2 - 1){{\left( {1.6263} \right)}^2} + (2 - 1){{\left( {2.3335} \right)}^2}}}{{8 - 4}}\)\( = 2.24\)

The ANOVA F statistic is the ratio of the MSTr and the MSE

\(F = \frac{{MS{T_r}}}{{MSE}} = \frac{{12851.3678}}{{6234.3995}} \approx 2.06\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table \(8\) containing the -value in the row\(d\,\,f\,n = I - 1 = {\rm{ }}4 - 1 = 3\)and\(d\,fd = N - I = {\rm{ 8}} - 4 = 4:\)

\(p > 0.10\)

If the P-value is less than the significance level, then reject the null hypothesis.

\(P{\rm{ }} > 0.05 \Rightarrow Fail\;to{\rm{ }}reject{\rm{ }}{H_0}\)

There is not sufficient evidence to support the claim that the true average form density is not the same for all manufacturers.

Hence,

There is not sufficient evidence to support the claim that the true average form density is not the same for all manufacturers.