Analysis Of Variance And Chi-Square Tests

Effects of Breakfast Drinks on Test Performance

For this question, we sought to test whether there breakfast drink differentially affect test performance. A researcher randomly assigns participants to five different ‘drink groups: Coffee, Tea, Orange Juice, Apple Juice, or Water. Each participant drinks their designated fluid for breakfast and then engages in a statistics exam. Higher scores on the exam indicate greater performance. The following data was collected:

Table 1: Data

Coffee	Tea	Orange Juice Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper	Apple Juice	Water
4	6	5	7	7
11	6	3	8	7
11	13	13	10	11
23	12	10	19	11
11	14	15	11	11

To test this, the following hypothesis was developed.

Null hypothesis (H₀): There is no significant difference in the test performance based on the type of breakfast drink taken.

Alternative hypothesis (H_A): There is significant difference in the test performance based on the type of breakfast drink taken.

Analysis of variance (ANOVA) test was performed and results compared at 5% level of significance. Results are presented below;

Table 2: ANOVA
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	26.960	4	6.740	.291	.880
Within Groups	462.800	20	23.140
Total	489.760	24

As can be seen in table 2 above, the p-value is given as 0.880 (a value greater than 5% level of significance), we therefore fail to reject the null hypothesis and conclude that there is no significant difference in the test performance based on the type of breakfast drink taken. That is, breakfast drink does not differentially affect test performance.

For this question, an investigator conducts an experiment to assess the effect of cellphone designs on ‘text talk’ (e.g., using k instead of okay, using idk instead of I don’t know). Three samples of individuals were given material to type on a particular cellphone type (a, b, or c), and the number of text talk uses by each participant was recorded (Anscombe, 2015). Higher scores indicate more text talk usage. The data are as follows:

Table 3: Data

A	B	C
0	6	6
4	8	5
0	5	9
1	4	4
0	2	6

To test this, the following hypothesis was developed.

Null hypothesis (H₀): There is no significant difference in the number of text talk based on the type cellphone.

Alternative hypothesis (H_A): There is significant difference in the number of text talk based on the type cellphone.

Analysis of variance (ANOVA) test was performed and results compared at 5% level of significance. Results are presented below;

Table 4: Descriptive

	N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for Mean
Lower Bound	Upper Bound
A	5	1.0000	1.73205	.77460	-1.1506	3.1506
B	5	5.0000	2.23607	1.00000	2.2236	7.7764
C	5	6.0000	1.87083	.83666	3.6771	8.3229
Total	15	4.0000	2.87849	.74322	2.4059	5.5941

Table 5: ANOVA
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	70.000	2	35.000	9.130	.004
Within Groups	46.000	12	3.833
Total	116.000	14

As can be seen in table 5 above, the p-value is given as 0.004 (a value less than 5% level of significance), we therefore reject the null hypothesis and conclude that there is significant difference in the number of text talk based on the type cellphone. That is, the type of cellphone design affects the text talk (Nikoli?, Muresan, Feng, & Singer, 2012).

Table 6: Multiple Comparisons
(I) Type of cellphone	(J) Type of cellphone	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
Lower Bound	Upper Bound
A	B	-4.00000*	1.23828	.022	-7.4418	-.5582
C	-5.00000*	1.23828	.005	-8.4418	-1.5582
B	A	4.00000*	1.23828	.022	.5582	7.4418
C	-1.00000	1.23828	1.000	-4.4418	2.4418
C	A	5.00000*	1.23828	.005	1.5582	8.4418
B	1.00000	1.23828	1.000	-2.4418	4.4418
*. The mean difference is significant at the 0.05 level.

Effects of Cellphone Design on Text Talk

 A post-hoc analysis by Bonferroni showed that there was significant difference in the number of text talk for cellphones A and B; A and C but no significant difference between B and C.

For this question, an investigator conducted an experiment involving the effects of three levels of caffeine on energy. Subjects are take part in each of three conditions. An additional cup of coffee is consumed in each condition. Energy is measured 10 minutes after each subject consumes the coffee. The following scores are recorded. The higher the score, the more energy

Table 7: Data

OneCup	TwoCups	ThreeCups
3	4	7
0	3	6
2	1	5
0	1	4
0	1	3

To test this, the following hypothesis was developed.

Null hypothesis (H₀): There is no significant difference in the amount of energy based on the level of caffeine.

Alternative hypothesis (H_A): There is significant difference in the amount of energy based on the level of caffeine.

Analysis of variance (ANOVA) test was performed and results compared at 5% level of significance. Results are presented below;

Table 8: Descriptive

	N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for Mean
Lower Bound	Upper Bound
One Cup	5	1.0000	1.41421	.63246	-.7560	2.7560
Two Cups	5	2.0000	1.41421	.63246	.2440	3.7560
Three Cups	5	5.0000	1.58114	.70711	3.0368	6.9632
Total	15	2.6667	2.22539	.57459	1.4343	3.8990

Table 9: ANOVA
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	43.333	2	21.667	10.000	.003
Within Groups	26.000	12	2.167
Total	69.333	14

As can be seen in table 9 above, the p-value is given as 0.003 (a value less than 5% level of significance), we therefore reject the null hypothesis and conclude that there is significant difference in the amount of energy based on the level of caffeine. That is, the level of caffeine affects the amount of energy (Tofallis, 2009).

Table 10: Multiple Comparisons
(I) Caffeine level	(J) Caffeine level	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
Lower Bound	Upper Bound
One Cup	Two Cups	-1.00000	.93095	.912	-3.5875	1.5875
Three Cups	-4.00000*	.93095	.003	-6.5875	-1.4125
Two Cups	One Cup	1.00000	.93095	.912	-1.5875	3.5875
Three Cups	-3.00000*	.93095	.022	-5.5875	-.4125
Three Cups	One Cup	4.00000*	.93095	.003	1.4125	6.5875
Two Cups	3.00000*	.93095	.022	.4125	5.5875
*. The mean difference is significant at the 0.05 level.

A post-hoc analysis by Bonferroni showed that there was significant difference in the amount of energy for one cup and three cups; two cups and three cups but no significant difference in the amount of energy between one cup and two cups.

Prior to an election, a survey was conducted to see if choice of political party depends on gender. The following data was collected from the survey:

Table 11: Data

	Liberal	Conservative
Female	118	82
Male	68	132

To test this, the following hypothesis was developed.

Null hypothesis (H₀): There is no significant association between political party one is and the gender of the person.

Alternative hypothesis (H_A): There is significant association between political party one is and the gender of the person.

Chi-Square test of independence was performed at 5% level of significance. Results are given below;

Table 12:Party * Gender Cross tabulation
	Gender	Total
Female	Male
Party	Conservative	82	132	214
Liberal	118	68	186
Total	200	200	400

Table 13: Chi-Square Tests
	Value	df	Asymp. Sig. (2-sided)	Exact Sig. (2-sided)	Exact Sig. (1-sided)
Pearson Chi-Square	25.123a	1	.000
Continuity Correctionb	24.128	1	.000
Likelihood Ratio	25.399	1	.000
Fisher’s Exact Test				.000	.000
N of Valid Cases	400
a. 0 cells (0.0%) have expected count less than 5. The minimum expected count is 93.00.
b. Computed only for a 2×2 table

As can be seen in table 13 above, the p-value is given as 0.000 (a value less than 5% level of significance), we therefore reject the null hypothesis and conclude that there is significant association between political party one is and the gender of the person. Majority of females associate with Liberal party while majority of males associate with Conservative party. 

Effects of Caffeine Levels on Energy

For this question, we want to know if there is a relationship between choice of party food and alcohol consumption. We survey individuals who drink alcohol at parties and those who do not as to their party food of choice. The data collected is below.

Table 14: Data

	Pizza	Nachos	Chicken Wings	Potato Chips
Drinkers	13	5	12	25
Non-Drinkers	19	15	28	18

To test this, the following hypothesis was developed.

Null hypothesis (H₀): There is no significant association between choice of party food and alcohol consumption.

Alternative hypothesis (H_A): There is significant association between choice of party food and alcohol consumption. Chi-Square test of independence was performed at 5% level of significance. Results are given below;

Table 15: Choice of party food * Alcohol consumption Cross tabulation
	Alcohol consumption	Total
Drinkers	Non-Drinkers
Choice of party food	Chicken Wings	12	28	40
Nachos	5	15	20
Pizza	13	19	32
Potato Chips	25	18	43
Total	55	80	135

Table 16: Chi-Square Tests
	Value	df	Asymp. Sig. (2-sided)
Pearson Chi-Square	9.356a	3	.025
Likelihood Ratio	9.435	3	.024
N of Valid Cases	135
a. 0 cells (0.0%) have expected count less than 5. The minimum expected count is 8.15.

As can be seen in table 16 above, the p-value is given as 0.000 (a value less than 5% level of significance), we therefore reject the null hypothesis and conclude that there significant association between choice of party food and alcohol consumption. Majority of non-drinkers prefer foods such as chicken wings, Nachos and Pizza while majority of drinkers prefer potato chips.

For question 8, a faculty member believes that students’ final grades are by both, their IQ, and the engagement they demonstrate in the class (via Blackboard checking).

Table 17: Data

Final Grade	Blackboard checking	IQ
91	60	134
70	22	107
84	41	120
56	10	102
49	2	105
78	33	110
94	35	105
68	20	100

This was tested using a regression analysis. The results are presented below; 

Table 18: Model Summary
Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.972a	.944	.922	4.49647
a. Predictors: (Constant), IQ, Blackboard checking

From table 18 above, we can see the value of R-Square to be 0.944; this implies that 94.4% of the variation in the dependent variable (final grades) is explained by the two independent variables (IQ and the engagement they demonstrate in the class) in the model.

Table 19: ANOVAa
Model	Sum of Squares	df	Mean Square	F	Sig.
1	Regression	1704.409	2	852.205	42.150	.001b
Residual	101.091	5	20.218
Total	1805.500	7
a. Dependent Variable: Final Grade
b. Predictors: (Constant), IQ, Blackboard checking

Looking at the ANOVA table (table 19 above), we can see that the p-value for the F-Statistics is 0.000 (a value less than 5% level of significance), we therefore reject the null hypothesis and conclude that the regression line significant at 5% level of significance.

Table 20: Regression Coefficients
Model	Unstandardized Coefficients	Standardized Coefficients	t	Sig.
B	Std. Error	Beta
1	(Constant)	139.219	27.253		5.108	.004
Blackboard checking	1.271	.174	1.456	7.303	.001
IQ	-.914	.283	-.645	-3.235	.023
a. Dependent Variable: Final Grade

Based on the results of table 20 above (regression coefficients), the regression equation for the model is given as follows; 

The constant variable (intercept) is given as 139.219; this means that holding all the other two factors constant (zero values for IQ and Blackboard checking) we would expect the average final grade to be 139.219.

The coefficient for the Blackboard checking is 1.271; this implies that a unit increase in the student’s engagement in the class would result to an increase in the final grade by 1.271. Similarly, a unit decrease in the student’s engagement in the class would result to a decrease in the final grade by 1.271.

The coefficient for the IQ is -0.914; this implies that a unit increase in the student’s IQ level would result to a decrease in the final grade by 0.914. Similarly, a unit decrease in the student’s IQ level would result to an increase in the final grade by 0.914.

For this question, a researcher wants to know if the memory technique used differentially effects recall of young and old participants. Participants are randomly assigned to one of five memory technique groups: Counting, Rhyming, Adjective, Imagery, and Intention. After training, participant’s recall is measured. Higher scores indicate greater memory. The data collected is as follows:

Table 21: Data

	Counting	Rhyming	Adjective	Imagery	Intention
Old	6	6	8	16	14
8	6	6	11	5
10	6	14	9	10
4	11	11	23	11
6	6	13	12	14
5	3	13	10	15
7	8	10	19	11
7	7	11	11	11
Young	8	10	14	20	21
6	7	11	16	19
4	8	18	16	17
6	10	14	15	15
7	4	13	8	22
6	7	22	16	16
5	10	17	20	22
7	6	16	22	22

To test this, the following hypothesis was developed.

Null hypothesis (H₀): There is no significant difference in the memory recall based on the different memory techniques.

Alternative hypothesis (H_A): There is significant difference in the memory recall based on the different memory techniques.

A two-way Analysis of variance (ANOVA) test was performed and results compared at 5% level of significance. Results are presented below; 

Table 22: Tests of Between-Subjects Effects
Source	Type III Sum of Squares	df	Mean Square	F	Sig.
Corrected Model	1600.263a	9	177.807	18.587	.000
Intercept	10511.113	1	10511.113	1098.791	.000
Memory_Technique	1220.825	4	305.206	31.905	.000
Age	208.013	1	208.013	21.745	.000
Memory_Technique * Age	171.425	4	42.856	4.480	.003
Error	669.625	70	9.566
Total	12781.000	80
Corrected Total	2269.888	79
a. R Squared = .705 (Adjusted R Squared = .667)

As can be seen in table 22 above, the p-value is given as 0.000 (a value less than 5% level of significance), we therefore reject the null hypothesis and conclude that there is significant difference in the memory recall based on the different memory techniques. That is, the type of memory technique used differentially effects recall of young and old participants.

References

Anscombe, F. J. (2015). The Validity of Comparative Experiments. Journal of the Royal Statistical Society, 111(3), 181–211.

Nikoli?, D., Muresan, R. C., Feng, W., & Singer, W. (2012). Scaled correlation analysis: a better way to compute a cross-correlogram. European Journal of Neuroscience, 35(5), 1–21. doi:10.1111/j.1460-9568.2011.07987.x

Tofallis, C. (2009). Least Squares Percentage Regression. Journal of Modern Applied Statistical Methods, 7(5), 526–534.