Recorded weight losses in lbs for 21 participants of three different diet programs wereDiet Program Weight Losses in lbs. x s#1 16 29 18 24 24 19 23 21.86 4.45#2 21 25 47 38 25 29 30 30.71 8.96#3 12 21 24 25 18 19 25 20.57 4.72(a) Run Hartley’s F-max test. Does evidence suggest the variances could be different at the α = 5% level of significance? Explain, and show any work you can.(b) The other ANOVA conditions (normality, independent random samples) were reasonably satisfied, so I ran an analysis of variance, and here is the incomplete table of output.Please fill in the missing values.Sum of Squares Degrees of Freedom Mean Square F Test StatisticSSB = 427.0 2 -------- -------------SSW =---------- ------------- --------- TSS = 1161.0 ------------(c) What is the p-value for this ANOVA test? Based on your p-value, is there significant evidence at the α = 5% level of significance to suggest the actual mean weight losses differ? Explain. **Recorded Weight Losses in lbs for 21 Participants of Three Different Diet Programs**| Diet Program | Weight Losses in lbs | $\bar{x}$ | $s$ ||--------------|----------------------|-------------|--------|| #1 | 16, 29, 18, 24, 24, 19, 23 | 21.86 | 4.45 || #2 | 21, 25, 47, 38, 25, 29, 30 | 30.71 | 8.96 || #3 | 12, 21, 24, 25, 18, 19, 25 | 20.57 | 4.72 |**(a)** Run Hartley’s $ F $-max test. Does evidence suggest the variances could be different at the $\alpha = 5\%$ level of significance? Explain, and show any work you can.**(b)** The other ANOVA conditions (normality, independent random samples) were reasonably satisfied, so I ran an analysis of variance, and here is the incomplete table of output. Please fill in the missing values.| Sum of Squares | Degrees of Freedom | Mean Square | $F$ Test Statistic ||----------------|--------------------|-------------|----------------------|| $SSB = 427.0$ | 2 | | || $SSW =$ | | | || $TSS = 1161.0$| | | |**(c)** What is the $ p $-value for this ANOVA test? Based on your $ p $-value, is there significant evidence at the $\alpha = 5\%$ level of significance to suggest the actual mean weight losses differ? Explain.

Number of diet programs is, k=3. Number of participants participate in the program is, n=21. The recorded weight of the participate is given as, Diet program Weight losses in lbs x¯ s #1 16 29 18 24 24 19 23 21.86 4.45 #2 21 25 47 38 25 29 30 30.71 8.96 #3 12 21 24 25 18 19 25 20.57 4.72a) The largest sample variance in the groups is s2max=8.96.The smallest sample variance in the groups is s2min=4.45. The expression for F-max test is,Fratio=s2maxs2min=8.964.45=2.01349 Compute degrees of freedom ( df ), based on the biggest sample size ( n ) in any groupdf=29-1=28 Based on the degrees of freedom ( df=28) and the number of groups ( k=3 ), the critical F value at 0.05 significance is 2.40. The Fratio is less than the given F-max value hence the hypothesis is accepted.b)The sum of square of total is, TSS=1161.0 The sum of square between the group is, SSB=427.0 The sum of square within the groups is, SSW=TSS-SSB=1161-427=734 Now, degree of freedom of groups is, Dfbetween groups =k-1=3-1=2 Degree of freedom of totals is, Dftotals=n-1=21-1=20 Degree of freedom within the groups is, Dfwithin the groups=n-k=21-3=18Now, mean square between the groups is, MSB=SSBDfbetween groups=4272=213.5 Mean square within the groups is, MSW=SSWDfwithin groups=73418=40.78 Then F-statistic test is,Fstat=MSBMSW=213.540.78=5.24 Hence, the required ANOVA table is, Sum of square Degree of freedom Mean square F-statistic test SSB=427 2 213.5 5.24 SSW=734 18 40.78 TSS=1161 20 c)The hypothesis that there is a significance difference between mean weight loss of participants in diet programs. Null and alternative Hypothesis are, H0:μ1=μ2=μ3=μ4H1:atleast one of the mean differ from otherThe level of significance is α=0.05 From the ANOVA table, The value of f-statistic test is 5.24.Degree of freedom of groups is 2Degree of freedom within the groups is 18. Hence, the p-value at 5.24,2,18 is 0.01616. So the null hypothesis is rejected and it can be accepted that 5% of significance can make the mean differ.

Answered: Recorded weight losses in lbs for 21…

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Recorded weight losses in lbs for 21 participants of three different diet programs were

Diet Program Weight Losses in lbs. x s
#1 16 29 18 24 24 19 23 21.86 4.45
#2 21 25 47 38 25 29 30 30.71 8.96
#3 12 21 24 25 18 19 25 20.57 4.72

(a) Run Hartley’s F-max test. Does evidence suggest the variances could be different at the α = 5% level of significance? Explain, and show any work you can.

(b) The other ANOVA conditions (normality, independent random samples) were reasonably satisfied, so I ran an analysis of variance, and here is the incomplete table of output.
Please fill in the missing values.
Sum of Squares Degrees of Freedom Mean Square F Test Statistic
SSB = 427.0 2 -------- -------------
SSW =---------- ------------- ---------
TSS = 1161.0 ------------

(c) What is the p-value for this ANOVA test? Based on your p-value, is there significant evidence at the α = 5% level of significance to suggest the actual mean weight losses differ? Explain.

**Recorded Weight Losses in lbs for 21 Participants of Three Different Diet Programs**

| Diet Program | Weight Losses in lbs | $\bar{x}$ | $s$ |
|--------------|----------------------|-------------|--------|
| #1 | 16, 29, 18, 24, 24, 19, 23 | 21.86 | 4.45 |
| #2 | 21, 25, 47, 38, 25, 29, 30 | 30.71 | 8.96 |
| #3 | 12, 21, 24, 25, 18, 19, 25 | 20.57 | 4.72 |

**(a)** Run Hartley’s $ F $-max test. Does evidence suggest the variances could be different at the $\alpha = 5\%$ level of significance? Explain, and show any work you can.

**(b)** The other ANOVA conditions (normality, independent random samples) were reasonably satisfied, so I ran an analysis of variance, and here is the incomplete table of output. Please fill in the missing values.

| Sum of Squares | Degrees of Freedom | Mean Square | $F$ Test Statistic |
|----------------|--------------------|-------------|----------------------|
| $SSB = 427.0$ | 2 | | |
| $SSW =$ | | | |
| $TSS = 1161.0$| | | |

**(c)** What is the $ p $-value for this ANOVA test? Based on your $ p $-value, is there significant evidence at the $\alpha = 5\%$ level of significance to suggest the actual mean weight losses differ? Explain.

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