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

Determined values are

a).$SSTR=46$.

b). $MSTR=15.33$.

c).$SSE=28$.

d).$MSE=2.33$.

e).$F-\text{statistic}=6.58$.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Part (a) Step 2: Explanation

Based on the information provided,

$k=4$

$n_1=3$

$n_2=5,$

$n_3=5,$

$n_4=3$

$n=3+5+5+3$

$n=16$

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

$\overline{x_1}=\frac{6+3+3}{3}$

$=4$

$\overline{x_2}=\frac{9+5+7+8+6}{5}$

$=7$

$\overline{x_3}=\frac{4+4+2+2+3}{5}$

$=3$

$\overline{x_2}=\frac{8+4+6}{3}$

$=6$

Part (a) Step 3: Explanation

Calculate the mean:

$\overline{x}=\frac{6+3+3+9+5+7+8+6+4+4+2+2+3+8+4+6}{16}$

$=5$

SSTR (sum of squares treatment):

$SSTR=n_1{\left(\overline{x_1}-\overline{x}\right)}^2+n_2{\left(\overline{x_2}-\overline{x}\right)}^2+n_3{\left(\overline{x_3}-\overline{x}\right)}^2+n_4{\left(\overline{x_4}-\overline{x}\right)}^2$

$=3(4-5{)}^2+5(7-5{)}^2+5(3-5{)}^2+3(6-5{)}^2$

$=46$

Part (b) Step 1: Given Information

Given data:

Part (b) Step 2: Explanation

We know,

$SSTR=46$

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

$=\frac{46}{4-1}$

$=15.33$

Part (c) Step 1: Given Information

Given data:

Part (c) Step 2: Explanation

Each sample's variance is $s_1^2=3,$

$s_2^2=2.5,$

$s_3^2=1,$

$s_4^2=4$

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

$=(3-1)\left(3\right)+(5-1)(2.5)+(5-1)\left(1\right)+(3-1)4$

$=28$

Part (d) Step 1: Given Information

Given data:

Part (d) Step 2: Explanation

We know

$n=16,$

$k=4$

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

$=\frac{28}{16-4}$

$=2.33$

Part (e) Step 1: Given Information

Given data:

Part (e) Step 2: Explanation

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

$=\frac{15.33}{2.33}$

$=6.58$