Statistics Exam Scores, Regression Analysis, And ANOVA: Solutions And Interpretations

Frequency Distribution of Examination Scores

1. The frequency distribution tables were constructed in MS Excel and have been provided below.

Table 1: Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper	FREQUENCY
50-60	50	59	54.5	3
60-70	60	69	64.5	2
70-80	70	79	74.5	5
80-90	80	89	84.5	4
90-100	90	99	94.5	6

Table 2: Cumulative Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	CUMULATIVE FREQUENCY
50-60	50	59	54.5	3	3
60-70	60	69	64.5	2	5
70-80	70	79	74.5	5	10
80-90	80	89	84.5	4	14
90-100	90	99	94.5	6	20

Table 3: Relative Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	RELATIVE FREQUENCY
50-60	50	59	54.5	3	0.15
60-70	60	69	64.5	2	0.10
70-80	70	79	74.5	5	0.25
80-90	80	89	84.5	4	0.20
90-100	90	99	94.5	6	0.30

Table 4: Cumulative Relative Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	RELATIVE FREQUENCY	CUMULATIVE RELATIVE FREQUENCY
50-60	50	59	54.5	3	0.15	0.15
60-70	60	69	64.5	2	0.10	0.25
70-80	70	79	74.5	5	0.25	0.50
80-90	80	89	84.5	4	0.20	0.70
90-100	90	99	94.5	6	0.30	1.00

Table 5: Percent Frequency Distribution of Examination Scores

CLASS INTERVAL	LOWER BOUND	UPPPER BOUND	CLASS MID-POINT	FREQUENCY	PERCENT FREQUENCY
50-60	50	59	54.5	3	15.00%
60-70	60	69	64.5	2	10.00%
70-80	70	79	74.5	5	25.00%
80-90	80	89	84.5	4	20.00%
90-100	90	99	94.5	6	30.00%

From percentage distribution of students based on their examination scores revealed that the nature of the histogram (percentage distribution graph). Accumulation of most of the students was observed above examination score of 70. Left skewness was evident from the shape of the histogram (Black, 2009). It can be inferred that students are getting very good marks in examination.

Answer 2

1. It is known that the “degrees of freedom” of residual in a regression model is taken as (n-p-1), where p is the degrees of freedom of the regression model. Here , so the value of n (sample size) is calculated as 41.
2. The margin of error at 5% level of significance is and the confidence interval of the slope is [0.029 – 0.041, 0.029 + 0.041] = [-0.012, 0.07]. The standard error of the slope indicates that the values of supply are very closely accumulated around the regression line of X, unit price (in thousands of dollars).
3. The given regression table is incomplete, where the values of SS for regression model and residue are provided. Now coefficient of determinant is known as , where SST = SSE + SSM = 7389.951. Hence, coefficient of determinant is calculated to be . This implies that 95.2% variation of supply (Y) is explained by the unit price (X).
4. Coefficient of correlation implies that the supply (Y) and unit price (X) are positively correlated. The value of the correlation indicated a steep increase in supply for change in unit price.
5. The regression equation is

Now for Unit Price = $ 50,000, the dependent variable is calculated as (in thousands of units). Hence predicted supply is approximately 55,530 units (Black, 2011). 

Answer 3

1. The ANOVA table is given in table 6 (constructed in MS Excel).

Table 6: ANOVA for programs in Allied Corporation

ANOVA: Single Factor
Groups	Count	Sum	Average	Variance
Program A	5	725	145	525
Program B	5	675	135	425
Program C	5	950	190	312.5
Program D	5	750	150	637.5
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	8750	3	2916.666667	6.1403	0.0055	3.2388
Within Groups	7600	16	475
Total	16350	19

The suggestion to Allied Corporation was provided based on the ANOVA results. The exploratory analysis shows that average output of day’s work for the C program is the maximum with average value of 190. The inferential analysis of ANOVA established that indeed, the output from C program is significantly (F = 6.14, p < 0.05) greater than that of from other three programs. Allied corporation was suggested to assign all of their employees in program C (Triola, 2013). 

Answer 4

1. The table 7 shows the regression output at 5% level of significance, calculated in Excel. The regression equation from the model is formed as,

Table 7: Regression Model for Weekly Sales and Price of Competitor’s Product at 5% Level of significance

SUMMARY OUTPUT	Regression 95%
Regression Statistics
Multiple R	0.8778
R Square	0.7706
Adjusted R Square	0.6558
Standard Error	1.8374
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	2.0000	45.3528	22.6764	6.7168	0.0526
Residual	4.0000	13.5043	3.3761
Total	6.0000	58.8571
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.5976	4.0522	0.8878	0.4248	-7.6532	14.8484
Price	41.3200	13.3374	3.0981	0.0363	4.2896	78.3505
Advertising	0.0132	0.3276	0.0404	0.9697	-0.8963	0.9228

The table 8 shows the regression output at 10% level of significance. The model is significant and that is inferred from the significance level (F = 6.72, p = 0.0526) of the model (Montgomery, Peck, and Vining, 2012).

Table 8: Regression Model for Weekly Sales and Price of Competitor’s Product at 10% Level of significance

SUMMARY OUTPUT	Regression 90%
Regression Statistics
Multiple R	0.8778
R Square	0.7706
Adjusted R Square	0.6558
Standard Error	1.8374
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	2.0000	45.3528	22.6764	6.7168	0.0526
Residual	4.0000	13.5043	3.3761
Total	6.0000	58.8571
	Coefficients	Standard Error	t Stat	P-value	Lower 90.0%	Upper 90.0%
Intercept	3.5976	4.0522	0.8878	0.4248	-5.0411	12.2364
Price	41.3200	13.3374	3.0981	0.0363	12.8868	69.7532
Advertising	0.0132	0.3276	0.0404	0.9697	-0.6851	0.7116

Individual relation of the independent variables reveal that advertisement expenditure (X2) is not significantly (t = 0.04, p =0.97) related to sales of the product (Y), whereas unit price of the competitors’ (X1) is significantly (t = 3.1, p < 0.1) related to sales of the product (Y) (Cox, and Wermuth, 2014).

1. The table 9 shows the regression output at 10% level of significance with unit price of competitors’ (X2) as the only variable (Vittinghoff et al., 2011). The new regression equation is calculated as

Table 9: Regression Model for Weekly Sales and Price of Competitor’s Product at 10% Level of significance with Single Factor

SUMMARY OUTPUT	Regression 90%
Regression Statistics
Multiple R	0.8778
R Square	0.7705
Adjusted R Square	0.7246
Standard Error	1.6438
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	1.0000	45.3473	45.3473	16.7831	0.0094
Residual	5.0000	13.5098	2.7020
Total	6.0000	58.8571
	Coefficients	Standard Error	t Stat	P-value	Lower 90.0%	Upper 90.0%
Intercept	3.5818	3.6082	0.9927	0.3664	-3.6889	10.8525
Price	41.6031	10.1552	4.0967	0.0094	21.1398	62.0663

Slope of the independent variable, unit price of competitors’ (X2) is 41.60 (t = 4.09, p < 0.05) that indicates existence of a positive correlation with the sales of the products (Y). Hence, from regression equation it can be said that sales of product (Y) increases by 41.60 units for one unit increase in competitors’ price of products (Burns, Bush, and Sinha, 2014).

References

Black, K., 2009. Business statistics: Contemporary decision making. John Wiley & Sons.

Black, K., 2011. Business statistics: for contemporary decision making. John Wiley & Sons.

Burns, A.C., Bush, R.F. and Sinha, N., 2014. Marketing research (Vol. 7). Harlow: Pearson.

Cox, D.R. and Wermuth, N., 2014. Multivariate dependencies: Models, analysis and interpretation. Chapman and Hall/CRC.

Montgomery, D.C., Peck, E.A. and Vining, G.G., 2012. Introduction to linear regression analysis (Vol. 821). John Wiley & Sons.

Triola, M.F., 2013. Elementary statistics using Excel. Pearson.

Vittinghoff, E., Glidden, D.V., Shiboski, S.C. and McCulloch, C.E., 2011. Regression methods in biostatistics: linear, logistic, survival, and repeated measures models. Springer Science & Business Media.