Question

SAMPLE 1	SAMPLE 2	SAMPLE 3
1	10	4
9	4	16
	8	10
	6
	2

Short answer

The proportion of explained variability (SSR) among overall variability is calculated using the sum of squares of total (SST), sum of squares due to regression (SSR), sum of squares of errors (SSE), and R-square (SST)

Step-by-step solution

Introduction.

Simple random sampling (SRS) is a method for selecting a sample of n number of sampling units from a population of N number of sampling units.

Given Information (part a).

We present data from separate simple random samples drawn from multiple populations; compute SST, SSTR, and SSE using the computational formulas in each case.

  Step 3: Explanation (part a).

Given Information (part b).

We give data from different simple random samples selected from multiple populations, and then compute SST, SSTR, and SSE using the appropriate computational methods.

  Step 5: Explanation (part b).

We have

$$\begin{gathered} k=3, \\ n\_1=2,n\_2=5,n\_3=3, \\ T\_1=10,T\_2=30,andT\_3=30 \\ n=sumn\_j=2+5+3=10 \\ sumx\_i=sumT\_j=10+30+30=70 \end{gathered}$$

Summing the squares of all the data in the above table yields

$\sum x_i^2=(1{)}^2+(9{)}^2+(10{)}^2+\dots .+(16{)}^2+(10{)}^2=674$

Given Information (part c).

We provide data from several basic random samples drawn from multiple populations, and then use the proper computational methods to compute SST, SSTR, and SSE.

  Step 7: Explanation (part c).

Consequently,

$$\begin{gathered} SST=(sum{x}_i{)}^2-(sum{x}_i{)}^2/n \\ =674-(70{)}^2/10 \\ =674-490=184 \\ SSTR=sum(T_j^2)/n-(sum{x}_j{)}^2/n \\ =(10{)}^2/2+(30{)}^2/5+(30{)}^2/3-(70{)}^2/10 \\ =530-490 \\ =40 \\ SSE=SST-SSTR \\ =184-40 \\ =144 \end{gathered}$$

Given Information (part d).

We offer data from numerous basic random samples selected from diverse populations, and then compute SST, SSTR, and SSE using the appropriate computational methods.

  Step 9: Explanation (part d).

Both the results are the same. Even though we use a different version of computations both yield the same results.

Given Information (part e).

We provide data from a variety of basic random samples drawn from various demographics, and then use the proper computational methods to compute SST, SSTR, and SSE.

  Step 11: Explanation (part e).

Thus treatment mean square is

$$\begin{gathered} MSTR=SSTR/k-1 \\ =40/3-1 \\ =20 \end{gathered}$$

The error mean square is

$$\begin{gathered} MSE=SSE/n-k \\ =144/10-3 \\ =20.57 \end{gathered}$$

The value of $F$-statistic is

$$\begin{gathered} F=MSTR/MSE \\ =20/2.57 \\ =0.97 \end{gathered}$$

Given Information (part f).

We supply data from a range of basic random samples selected from diverse demographics, and then compute SST, SSTR, and SSE using the appropriate computational methods.

  Step 13: Explanation (part f).

Given Information (part g).

We provide data from a variety of basic random samples drawn from various demographics, and then use the proper computational methods to compute SST, SSTR, and SSE.

  Step 15: Explanation (part g).

The null and alternative hypotheses are

$H_0:{\mu }_1={\mu }_2={\mu }_3$

$H_1$: Not all the means are equal

We are to perform the test at the $5\%$significance level; so $\alpha =0.05$

We have 3 populations under consideration, or $k=3$, and that the number of observations total 10 , or $n=10$.

Hence the degrees of freedom for the $F$-statistic is

$$\begin{gathered} df=(k-1,n-k) \\ =(3-1,10-3) \\ =(2,7) \end{gathered}$$

From table VIII, the critical value at the $5\%$level of significance is $F_{0.05}=4.74$

Referring to table Vill with$df=(2,7)$, we find $P>0.10$

Because the P-value is greater than the significance level we do not reject $H_0$

The data do not provide sufficient evidence to conclude that the means of the populations from which the samples were drawn are not all the same.