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One-Way ANOVA ("analysis of variance") examines the means of two or more independent groups to see if the related population means differ statistically.

Consider the following table and use the formula to find the mean of all observations:

$\begin{array}{|lll|}\hline n_j& {\stackrel{¯}{x}}_j& s_j\\ 39& 4.49& 0.80\\ 20& 5.06& 0.79\\ 28& 5.75& 1.20\\ 14& 5.99& 0.97\\ \hline\end{array}$

Using the following formula, calculate the mean of all observations, $\stackrel{¯}{x}$, using the provided data.

$\stackrel{¯}{x}=\frac{n_1\stackrel{¯}{x_1}+n_2\stackrel{¯}{x_2}+….+n_k\stackrel{¯}{{\stackrel{¯}{k}}_k}}{n_1+n_2….+n_k}$

Where $k$represents the number of people in the study, $n$represents the total number of observations, and $\stackrel{¯}{x}$represents the overall mean.

Now, using the values from the provided data,

$$\begin{gathered} \stackrel{¯}{x}=\frac{39.4.49+20.5.06+28.5.75+14.5.99}{39+20+28+14} \\ =5.16 \end{gathered}$$

One-Way ANOVA ("analysis of variance") examines the means of two or more independent groups to see if the related population means differ statistically.

Assume that all four populations have sample sizes of $14$. Calculate the mean of all observations using the formula:

$$\begin{gathered} \stackrel{¯}{x}=\frac{14.4.49+14.5.06+14.5.75+14.5.99}{14+14+14+14} \\ =5.32 \end{gathered}$$

The sample means' mean is as follows:

$$\begin{gathered} \stackrel{¯}{x}=\frac{4.49+5.06+5.75+5.99}{4} \\ =5.32 \end{gathered}$$

As a result, the two estimated means are equivalent.

One-Way ANOVA ("analysis of variance") examines the means of two or more independent groups to see if the related population means differ statistically.

Calculate the F-statistic from the summary statistic by following the procedures below.

Using the formula below, find the mean of all observations, $\stackrel{¯}{x}$, for the given data.

$\stackrel{¯}{x}=\frac{n_1{\stackrel{¯}{x}}_1+n_2{\stackrel{¯}{x}}_2+……+n_k{\stackrel{¯}{x}}_k}{n_1+n_3+…..+n_k}$

Where $k$denotes the number of populations to be considered. The total number of observations is $n$, and the overall mean is $\stackrel{¯}{x}$.

Calculate the three sums of squares using the formula below: SSTR (treatment sum of squares), SSE (error sum of squares), and SST (total sum of squares).

$SSTR=n_j{\left({\stackrel{¯}{x}}_j-\stackrel{¯}{x}\right)}^2$

$SSE=\left(n_j-1\right)s_j^2$

$SST=SSTR+SSE$

Where $n_j$is the sample size from population $j$, ${\stackrel{¯}{x}}_j$is the sample mean from population $j$, and $s_j^2$is the sample variance from population $j$.

Obtain MSTR (treatment mean square) and MSE (error mean square).

$MSTR=\frac{SSTR}{k-1}$

and

$MSE=\frac{SSE}{n-k}$

The number of observations is $n$, and the number of populations is $k$.