Question

On page 539, we discussed how to use summary statistics (sample sizes, sample means, and sample standard deviations) to conduct a one-way ANOVA.

a. Verify the formula presented there for obtaining the mean of all the observations, namely,

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2+\cdots +n_k{\overline{x}}_k}{n_1+n_2+\cdots +n_k}.$

b. Show that, if all the sample sizes are equal, then the mean of all the observations is just the mean of the sample means.

c. Explain in detail how to obtain the value of the F-statistic from the summary statistics.

Short answer

(a) The mean of all the observations is$\overline{x}=5.16$.

(b) It is demonstrated that the mean of all observations equals the sample size mean of $5.32$.

(c) The F-statistic from the summary statistics has the value $F=\frac{MSTR}{MSE}$.

Step-by-step solution

Part(a) Step 1: Given Information

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

Part(a) Step 2: Explanation

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

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

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

$\overline{x}=\frac{n_1\overline{x_1}+n_2\overline{x_2}+\dots .+n_k\overline{{\overline{k}}_k}}{n_1+n_2\dots .+n_k}$

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

Now, using the values from the provided data,

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

Part(b) Step 1: Given Information

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

Part(b) Step 2: Explanation

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

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

The sample means' mean is as follows:

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

As a result, the two estimated means are equivalent.

Part(c) Step 1: Given Information

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

Part(c) Step 2: Explanation

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

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

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2+\dots \dots +n_k{\overline{x}}_k}{n_1+n_3+\dots ..+n_k}$

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

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).

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

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

$SST=SSTR+SSE$

Where $n_j$is the sample size from population $j$, ${\overline{x}}_j$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 $k$.