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).$SSTR=24$.

b).$MSTR=12$.

c).$SSE=16$.

d).$MSE=2.29$.

e).$F-\text{statistic}=5.25$.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Part (a) Step 2: Explanation

Based on the information provided,

$k=3$

$n_1=3,$

$n_2=3,$

$n_3=4$

$n=3+3+4$

$n=10$

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

$\overline{x_1}=\frac{8+4+6}{3}$

$=6$

$\overline{x_2}=\frac{2+1+3}{3}$

$=2$

$\overline{x_3}=\frac{4+3+6+3}{4}$

$=4$

Part (a) Step 3:Explanation

Calculate the mean.

$\overline{x}=\frac{8+4+6+2+!+3+4+3+6+3}{10}$

$=4$

SSTR (sum of squares treatment)

localid="1652767430243" $SSTR=n_1{\left(\overline{x_1}-\overline{x}\right)}^2+n_2{\left(\overline{x_2}-\overline{x}\right)}^2+n_3{\left(\overline{x_3}-\overline{x}\right)}^2$ $=3(6-4{)}^2+3(2-4{)}^2+4(4-4{)}^2$

$=24$

Part (b) Step 1: Given Information

Given data,

Part (b) Step 2: Explanation

We know

$SSTR=24$

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

$=\frac{24}{3-1}$

$=12$

Part (c) Step 1: Given Information

Given data:

Part (c) Step 2: Explanation

Each sample's variance is $s_1^2=4,$

$s_2^2=1,$

$s_3^2=2$

$SSE=\left(n_1-1\right)s_1^2+\left(n_2-1\right)s_2^2+\left(n_3-1\right)s_3^2$ $=(3-1)\left(4\right)+(3-1)\left(1\right)+(4-1)\left(2\right)$

$=16$

Part (d) Step 1: Given Information

Given data,

Part (d) Step 2: Explanation

We know

$n=10,$

$k=3$

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

$=\frac{16}{10-3}$

$=2.29$

Part (e) Step 1: Given Information

Given data:

Part (e) Step 2: Explanation

We know

$n=10,$

$k=3$

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

$=\frac{12}{2.29}$

$=5.25$