Question

Sample 1	Sample 2	Sample 3	Sample 4
6	9	4	8
3	5	4	4
3	7	2	6
	8	2
	6	3

Short answer

• The sum of squares of total (SST), sum of squares due to regression (SSR), sum of squares of errors (SSE), and R-square are used to quantify the fraction of explained variability (SSR) within overall variability (SST)

Step-by-step solution

Introduction.

• A method of selecting a sample of n number of sampling units from a population of N number of sampling units is known as simple random sampling (SRS).

Given Information (part a).

• 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 3: Explanation (part a).

Given Information (part b).

• 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 5: Explanation (part b).

We have

$$\begin{gathered} k=4, \\ {n}_1=3,n_2=5,n_3=5,n_4=3, \\ {T}_1=12,T_2=35,T_3=15and{T}_4=18 \\ n=sum{n}_j=3+5+5+3=16 \\ sum{x}_i=sum{T}_j=12+35+15+18=80 \end{gathered}$$

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

$\sum x_i^2=(6{)}^2+(3{)}^2+(3{)}^2+\dots .+(4{)}^2+(6{)}^2=438$

Given Information (part c).

• 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 7: Explanation (part c).

$$\begin{gathered} SST=(sum{x}_i{)}^2-(sum{x}_i{)}^2/n \\ =474-(80{)}^2/16 \\ =474-400 \\ =74 \\ SSTR=sum(T_j^2)/n-(sum{x}_j{)}^2/n \\ =(12{)}^2/3+(35{)}^2/5+(15{)}^2/5+(80{)}^2/16 \\ =446-400 \\ =46 \\ SSE=SST-SSTR \\ =74-46 \\ =28 \end{gathered}$$

Given Information (part d).

• 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 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 employ data from a range of basic random samples drawn from diverse demographics to compute SST, SSTR, and SSE, and then apply the appropriate computational methods.

  Step 11: Explanation (part e).

Thus treatment mean square is

$$\begin{gathered} MSTR=SSTR/k-1 \\ =46/4-1 \\ =15.33 \end{gathered}$$

The error mean square

$$\begin{gathered} MSE=SSE/n-k \\ =28/16-4 \\ =2.33 \end{gathered}$$

The value of $f$- statistic is

$$\begin{gathered} F=MSTR/MSE \\ =15.33/2.33 \\ =6.58 \end{gathered}$$

Given Information (part f).

• 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 13: Explanation (part f).

Given Information (part g).

• We use data from a range of basic random samples selected from diverse demographics to compute SST, SSTR, and SSE.

  Step 15: Explanation (part g).

The null and alternative hypotheses are

$H_0:{\mu }_1={\mu }_2={\mu }_3={\mu }_4$

$H_1$: Not all the means are equal

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

We have 4 populations under consideration, or $k=4$, and that the number of observations total 16 , or$n=16$.

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

$$\begin{gathered} df=(k-1,n-k) \\ =(4-1,16-4) \\ =(3,12) \end{gathered}$$

From table VIII, the critical value at the $5\%$level of significance is $F_{0.05}=3.49$

Referring to table VIII with $\mathrm{df}=(3,12)$, we find $0.005<P<0.01$

Because the P-value is smaller than the significance level we reject $H_0$

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