Question

We have provided data from independent simple random samples from several populations. In each case, determine the following items.

a. SSTR

b. MSTR

c. SSE

d. MSE

e. F

Short answer

The determined values are,

A).$SSTR=138$.

B).$MSTR=46$.

C).$SSE=72$.

D).$MSE=9$.

E).$F-\text{statistic}=5.11$.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Part (a) Step 2: Explanation

Based on the information provided,

$k=4$

$n_1=n_2=n_3=n_4=3$

$n=12$

Samples one, two, and three have the following mean values:

$\overline{x_1}=8,$

$\overline{x_2}=5,$

$\overline{x_3}=12,$

$\overline{x_4}=3$

Part (a) Step 3: Explanation

Calculate the mean:

$\overline{x}=\frac{84}{12}$

$=7$

SSTR (sum of squares treatment)

$SSTR=n_1{\left(\overline{x_1}-\overline{x}\right)}^2+n_2{\left(\overline{x_2}-\overline{x}\right)}^2+n_3{\left(\overline{x_3}-\overline{x}\right)}^2+n_4{\left(\overline{x_4}-\overline{x}\right)}^2+n_5{\left(\overline{x_5}-\overline{x}\right)}^2$ $=3(8-7{)}^2+3(5-7{)}^2+3(12-7{)}^2+3(3-7{)}^2$

$=138$

Part (b) Step 1: Given Information

Given data;

Part (b) Step 2: Explanation

We know $SSTR=138$

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

$=\frac{138}{4-1}$

$=46$

Part (c) Step 1: Given Information

Given data:

Part (c) Step 2: Given Information

Each sample's variance is $s_1^2=7,$

$s_2^2=13,$

$s_3^2=12,$

$s_4^2=4$

$SSE=\left(n_1-1\right)s_1{}^2+\left(n_2-1\right)s_2{}^2+\left(n_3-1\right)s_3{}^2$ $=(4-1)\left(7\right)+(4-1)\left(13\right)+(4-1)\left(12\right)+(4-1)4$

$=72$

Part (d) Step 1: Given Information

Given data:

Part (d) Step 2: Explanation

We know

$n=12,$

$k=4$

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

$=\frac{72}{12-4}$

$=9$

Part (e) Step 1: Given Information

Given data:

Part (e) Step 2: Explanation

To find the value of

$F-\text{statistic=}\frac{\text{MSTR}}{MSE}$

$=\frac{46}{9}$

$=5.11$