Frequency Distribution for Examination Scores of Students
1A Table 1: Frequency Distribution of Scores of Students
|
Class Interval |
Lower Bound |
Upper Bound |
Mid-Point |
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 Scores of Students
|
Class Interval |
Lower Bound |
Upper Bound |
Mid-Point |
Frequency |
|
50-60 |
50 |
59 |
54.5 |
3 |
|
60-70 |
60 |
69 |
64.5 |
5 |
|
70-80 |
70 |
79 |
74.5 |
10 |
|
80-90 |
80 |
89 |
84.5 |
14 |
|
90-100 |
90 |
99 |
94.5 |
20 |
Table 3: Relative Frequency Distribution of Scores of Students
|
Class Interval |
Lower Bound |
Upper Bound |
Mid-Point |
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 4: Cumulative Relative Frequency Distribution of Scores of Students
|
Class Interval |
Lower Bound |
Upper Bound |
Mid-Point |
Frequency |
|
50-60 |
50 |
59 |
54.5 |
0.15 |
|
60-70 |
60 |
69 |
64.5 |
0.25 |
|
70-80 |
70 |
79 |
74.5 |
0.50 |
|
80-90 |
80 |
89 |
84.5 |
0.70 |
|
90-100 |
90 |
99 |
94.5 |
1.00 |
Table 5: Percent Frequency Distribution of Scores of Students
|
Class Interval |
Lower Bound |
Upper Bound |
Mid-Point |
Frequency |
|
50-60 |
50 |
59 |
54.5 |
15.00% |
|
60-70 |
60 |
69 |
64.5 |
10.00% |
|
70-80 |
70 |
79 |
74.5 |
25.00% |
|
80-90 |
80 |
89 |
84.5 |
20.00% |
|
90-100 |
90 |
99 |
94.5 |
30.00% |
1B
Figure 1: Distribution of Scores of the Students
The histogram shows that the distribution of scores of the students is left skewed in nature. The conclusion may be drawn that examination scores of majority of the students are highly satisfactory (Hinton, 2014).
2A. The degree of freedom for residual is always sample size minus two. Hence sample size is (39+2) = 41 for the current problem.
2B. Standard error of unit price indicated that about 95% of the data should belong within 2*standard error (SE) of the estimate. For the current problem, it is observed that about 95% of the data falls within 0.042% of the fitted line of regression model. Hence, supply (demand) has a strong linear relation with unit price with low gradient.
2C. From the regression model SSM (sum square model) =354.689, SSE (sum square error) = 7035.262, hence, SST (sum square total) = SSE + SSM = 354.689 +7035.262 = 7389.951.
Coefficient of determination implied that the model is able to define 95.2% variation in supply. Therefore, it is possible to conclude that unit price is able to explain the variation in supply up to 95.2%.
2D. Coefficient of correlation between supply and unit price is . The correlation is highly positive and almost perfect in nature. For one unit increase in price the supply would also increase (Winston, W., 2016).
2E. Equation of the regression line is, where supply Y is in thousands of units, and unit price X is in thousands of dollars. Hence, for unit price of $50,000, the supply would be calculated as, thousand units.
3A. Table 6: ANOVA Single Factor Summary
|
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 |
2916.67 |
6.14 |
0.0056 |
3.24 |
|
Within Groups |
7600 |
16 |
475 |
|||
|
Total |
16350 |
19 |
3B. The average productivity from program C (M = 190) is higher than the average productivity either from program A (M = 145), program B (M = 135), or program D (M = 150). According to the ANOVA results the difference is significantly important (F = 6.14, p < 0.05). The calculated F value is greater than F critical value at 5% level of significance, and the null hypothesis that assumed equal productivity from the four programs is rejected. The Allied Corporation will be able to greatly increase productivity of its employees by enrolling employees in program C, compared to other three programs (Quirk, 2015).
4A. Table 7: SUMMARY OUTPUT for Regression Model (CI = 95%)
|
Regression Statistics |
||||||
|
Multiple R |
0.88 |
|||||
|
R Square |
0.77 |
|||||
|
Adjusted R Square |
0.66 |
|||||
|
Standard Error |
1.84 |
|||||
|
Observations |
7 |
|||||
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
2 |
45.35 |
22.68 |
6.72 |
0.053 |
|
|
Residual |
4 |
13.50 |
3.38 |
|||
|
Total |
6 |
58.86 |
||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
|
|
Intercept |
3.60 |
4.05 |
0.89 |
0.42 |
-7.65 |
14.85 |
|
Price |
41.32 |
13.34 |
3.10 |
0.04 |
4.29 |
78.35 |
|
Advertising |
0.01 |
0.33 |
0.04 |
0.97 |
-0.90 |
0.92 |
Estimated regression equation is Y (Sales) = 0.1 Advertising + 41.32 Price +3.60
4B. Table 8: SUMMARY OUTPUT Regression Model (90%) WITH IV: Price, Advertising
|
Regression Statistics |
||||||
|
Multiple R |
0.88 |
|||||
|
R Square |
0.77 |
|||||
|
Adjusted R Square |
0.66 |
|||||
|
Standard Error |
1.84 |
|||||
|
Observations |
7 |
|||||
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
2 |
45.35 |
22.68 |
6.72 |
0.053 |
|
|
Residual |
4 |
13.50 |
3.38 |
|||
|
Total |
6 |
58.86 |
||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 90.0% |
Upper 90.0% |
|
|
Intercept |
3.60 |
4.05 |
0.89 |
0.42 |
-5.04 |
12.24 |
|
Price |
41.32 |
13.34 |
3.10 |
0.04 |
12.89 |
69.75 |
|
Advertising |
0.01 |
0.33 |
0.04 |
0.97 |
-0.69 |
0.71 |
The regression model at α = 0.10 is significant. The implication of calculated F value of the model (F = 6.72, p = 0.053) is statistically significant.
4c. Competitor’s price is significantly related to sales (t = 3.10, p < 0.1), but advertising is not significantly related (t = 0.04, p = 0.97) to sales.
4d. Table 9: SUMMARY OUTPUT (90%) with Price as Independent Variable
|
Regression Statistics |
||||||
|
Multiple R |
0.88 |
|||||
|
R Square |
0.77 |
|||||
|
Adjusted R Square |
0.72 |
|||||
|
Standard Error |
1.64 |
|||||
|
Observations |
7 |
|||||
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
1 |
45.35 |
45.35 |
16.78 |
0.01 |
|
|
Residual |
5 |
13.51 |
2.70 |
|||
|
Total |
6 |
58.86 |
||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 90.0% |
Upper 90.0% |
|
|
Intercept |
3.58 |
3.61 |
0.99 |
0.37 |
-3.69 |
10.85 |
|
Price |
41.60 |
10.16 |
4.10 |
0.01 |
21.14 |
62.07 |
The independent factor, advertising expenditure is dropped from the model, and the regression model is re-estimated. New estimated regression equation is. Y (Sales) = 41.60 Prices
4e. Slope coefficient of price-sales model implies a positive and highly sensitive dependence on the independent factor. The weekly sales of the company’s product increase rapidly when the competitor raises unit price of its product. The slope coefficient of unit price of competitor is statistically significant (t = 4.10, p < 0.1) in predicting sales of the company (Schroeder, Sjoquist, and Stephan, 2016).
Reference
Hinton, P.R., 2014. Statistics explained. Routledge.
Quirk, T.J., 2015. One-way analysis of variance (ANOVA). In Excel 2013 for Social Sciences Statistics(pp. 177-196). Springer, Cham.
Schroeder, L.D., Sjoquist, D.L. and Stephan, P.E., 2016. Understanding regression analysis: An introductory guide (Vol. 57). Sage Publications.
Winston, W., 2016. Microsoft Excel data analysis and business modeling. Microsoft press.
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