Question

Create an ANOVA and regresion problem, show all steps and result.

Answer 1

1)

ANOVA Problem

Research Question: Do the teenagers spend equal time in playing video games on weekends across the three groups?

Null Hypothesis

H₀: µ_{Group 1} = µ_{Group 2} = µ_{Group 3} (Teenagers spend equal time in playing video games on weekends across the three groups)

Alternate Hypothesis

H_a: Not all Means are equal

Data Given

	10-12 Yrs Old	13-15 Yrs Old	16-18 Yrs Old
	Group 1	Group 2	Group 3	Overall
Mean	2	5	6	4.38
Variance	10	25	20
Std Dev	3.16	5.00	4.47
Sample Size	39	43	41	123

Overall Mean = (2*39+5*43+6*41)/(39+43+41) = 4.38

alpha = 0.05

Degress of Freedom:

df_Between = a – 1 = 3-1 =2

df_Within = N-a = 123-3 = 120

df_Total = N-1 = 123-1 = 122

Critical Values:

Time (df_Between, df_Within): (2,120) = 3.09

Decision Rule:

If F is greater than 3.09, reject the null hypothesis

Test Statistics:

Between Group Variance(SSB) = ∑n(Group Average – Overall Average)² = 39*(2 – 4.38)² + 43*(5 – 4.38)² + 41*(6 – 4.38)² = 345.04

Standard Deviation Pooled (s_p) = { [SD₁²*n₁+SD₂²*n₂+SD₃²*n₃] / [n₁+n₂+n₃] }^0.5 = 4.31

Within Group Variance = SSE

s_p = (SSE/(n-1))^1/2

where n = (39+43+41)/3 = 41

SSE = 743.09

SS_Betwen = 345.04

SS_Within = 743.09

SS_Total = SS_{Betwen ­­}+ SS_Within = 1088.13

MS = SS/df

               

F = MS_effect / MS_error

Hence,

F = 172.52/6.19 = 27.86

	SS	df	MS	F
Between	345.04	2	172.52	27.86
Within	743.09	120	6.19
Total	1,088.13	122

Result:

Our F = 27.86, we reject the null hypothesis

Conclusion:

Not all means are equal.

2)

Regression Problem:

Suppose we want to find the Ice cream consumption on the basis of Temperature (in Fahrenheit).

Data Set

temp	icecream
41	0.386
56	0.374
63	0.393
68	0.425
69	0.406
65	0.344
61	0.327
47	0.288
32	0.269
24	0.256
28	0.286
26	0.298
32	0.329
40	0.318
55	0.381
63	0.381
72	0.47
72	0.443
67	0.386
60	0.342
44	0.319
40	0.307
32	0.284
27	0.326
28	0.309
33	0.359
41	0.376
52	0.416
64	0.437
71	0.548

Solution:

Dependent Variable: Icecream

Independent Variable: Temperature

We expect the relationship to be positive as the temperature increases, people will spend more on ice-cream.

Steps for Regression in Excel

• Import the data in Excel
• In Data Analysis tool from Data Tab.
• Click on regression and select the Y values and X values. Select Labels and click OK.

Regression Output

Regression Statistics
Multiple R	0.7756
R Square	0.6016
Adjusted R Square	0.5874
Standard Error	0.0423
Observations	30
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.08	0.08	42.28	0.00
Residual	28	0.05	0.00
Total	29	0.13
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	0.21	0.02	8.37	0.00	0.16	0.26
temp	0.003	0.00	6.50	0.00	0.00	0.00

Since the Significance F Statistic is less than 0.05, we reject the null hypothesis ie the above model is significant.

Also, the p-value of the variable is less than 0.05, hence the variable is also significant in explaining the variation of dependent variable.

Regression Equation

Icecream = 0.21 + 0.003 temp