Question

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

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

Given Information (part a).

• We give data from different simple random samples selected from multiple populations, and we 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.

Explanation (part b). 

We have

$$\begin{gathered} k=5, \\ {n}_1=4,n_2=3,n_3=5,n_3=5,n_4=5,n_5=3 \\ {T}_1=20,T_2=18,T_3=25,T_4=30and{T}_5=27 \\ n=sum{n}_j=4+3+5+5+3=20 \\ sum{x}_i=sum{T}_j=20+18+30+25+27=120 \end{gathered}$$

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

$\sum x_i^2=(7{)}^2+(4{)}^2+(5{)}^2+\dots .+(9{)}^2+(11{)}^2=808$

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). 

$$\begin{gathered} SST=(sum{x}_i{)}^2-(sum{x}_i{)}^2/n \\ =808-(120{)}^2/20 \\ =808-720 \\ =88 \\ SSTR=sum(T_j^2)/n-(sum{x}_j{)}^2/n \\ =(20{)}^2/4+(18{)}^2/3+(30{)}^2/5+(25{)}^2/5+(27{)}^2/3+(120{)}^2/20 \\ =756-720 \\ =36 \\ SSE=SST-SSTR \\ =88-36 \\ =52 \end{gathered}$$

Given Information (part d). 

• 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 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 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 11: Explanation (part e). 

Thus treatment mean square is

$$\begin{gathered} MSTR=SSTR/k-1 \\ =36/5-1 \\ =9 \end{gathered}$$

The error mean square is

$$\begin{gathered} MSE=SSE/n-k \\ =52/20-5 \\ =2.33 \end{gathered}$$

The value of -statistic is

$$\begin{gathered} =MSTR/MSE \\ =9/3.47 \\ =2.60 \end{gathered}$$

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.

Explanation (part f). 

Given Information (part g). 

• 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 15: Explanation (part g). 

The null and alternative hypotheses are

$H_0:{\mu }_1={\mu }_2={\mu }_3={\mu }_4={\mu }_5$

$H_1$: Not all the means are equal

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

We have 5 populations under consideration, or $k=5$, and that the number of observations total 20 , or $n=20$.

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

$$\begin{gathered} df=(k-1,n-k) \\ =(5-1,20-5) \\ =(4,15) \end{gathered}$$

From table VIII, the critical value at the $5\%$level of significance is $F_{0.0s}=3.06$

Referring to table VIII with df $=(4,15)$, we find $0.05<P<0.10$

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.