Frequency Distribution of Students’ Scores
Score of students in an exam have been given as follows:
|
52 |
99 |
92 |
86 |
84 |
|
63 |
72 |
76 |
95 |
88 |
|
92 |
58 |
65 |
79 |
80 |
|
90 |
75 |
74 |
56 |
99 |
Table 1: Scores
The frequency distribution of the scores of the students is giving in the following table. The table also contains the cumulative frequency, the relative frequency, the cumulative relative frequency and the frequency in percentage or the percentage frequency. The scores have been divided in 6 intervals of width equal to 10 each.
|
Score |
Frequency |
Cumulative Frequency |
Relative Frequency |
Cumulative relative frequency |
Percentage frequency |
|
52-61 |
3 |
3 |
0.15 |
0.15 |
15.00% |
|
62-71 |
2 |
5 |
0.10 |
0.25 |
10.00% |
|
72-81 |
6 |
11 |
0.30 |
0.55 |
30.00% |
|
82-91 |
4 |
15 |
0.20 |
0.75 |
20.00% |
|
92-101 |
5 |
20 |
0.25 |
1.00 |
25.00% |
|
Grand Total |
20 |
100.00% |
100.00% |
Table 2: Frequency Distribution
the histogram of the scores of the students. The primary vertical axis represents the relative frequency and the secondary vertical axis represents the cumulative relative frequency.
The distribution seems to have higher frequency on the right hand side and is thus not symmetric. The distribution seems to be negatively skewed or right skewed. The interval 72 to 81 is seen to be the class interval which contains the median. It also is the class which has highest frequency and it is therefore the modal class.
The output of a regression analysis was given as given in the table below where the dependent variable, y is in thousands of unit and the independent variable X is in thousands of dollars.
|
ANOVA TABLE |
|||
|
Degrees of freedom |
Sum of Squares(SS) |
||
|
Regression |
1 |
354.689 |
|
|
Residual |
39 |
7035.262 |
|
|
Est.Coefficients |
Standard Error (SE) |
||
|
Intercept |
54.076 |
2.358 |
|
|
X |
0.029 |
0.021 |
Table 3: Regression Output
Then the sample size n can be computed by looking at the degrees of freedom of Regression and Residual. The degrees of freedom of the regression is k-1= 1 since there is only one independent variable and the degrees of freedom of residual is n-k-1= 39 (Given). Then n= 39+k +1= 39 + 1 = 40. So there are 40 observations in the sample.
The regression coefficient of X or the unit price has been given as 0.029 and the standard error as 0.021. Then the t-statistic to test for significance of the regression coefficient is given by their ratio which is equal to 0.029/0.021 which equals 1.38 and the critical t-statistic with alpha 0.05 and degree of freedom 1 was found to be 0.968. Then the observed statistic has a value greater than the critical value. Therefore the variable X or the price in units is inferred to be significant and hence the supply, y is related to X.
The coefficient of determination is the R-squared statistic of the regression model. It is the ratio of the explained variation by the model to the total variation and serves as a measure of goodness of fit for the model. The statistic is given by the ratio: SSR/ (SSR +SSE)
Regression Analysis Output
Here SSR is given to be 354.689 and SSE is given to be 7035.26.
Then the coefficient of determination, r2 = = 0.047996. This means that the model explains only 4.79% of the total variation of the dependent variable, supply (y).
The coefficient of correlation, denoted by r is given by the square root of the coefficient of determination, r2. Hence it is obtained as r = = = 0.21908. So the correlation coefficient between the supply in thousand units and price in thousand dollars is given by 0.21908.
The supply as predicted by the fitted regression model for the price equal to $50,000 is obtained by plugging in X = 50,000 in the regression equation: y= 54.076 + 0.029 X.
Then y = 54.076 + 0.029 x 50000 = 1504.076 units
In order to aid the Allied Corporation in increasing their productivity in terms of the output of the line workers, four programs, namely A, B, C and D were designed. Twenty employees were randomly selected and assigned to any one of the four programs and their daily output were compared with one another using ANOVA to check whether one program performed better in bettering productivity than the others or not. The following table shows the observed output for each group.
|
Program A |
Program B |
Program C |
Program D |
|
150 |
150 |
185 |
175 |
|
130 |
120 |
220 |
150 |
|
120 |
135 |
190 |
120 |
|
180 |
160 |
180 |
130 |
|
145 |
110 |
175 |
175 |
Table 4: Line worker output for Groups A, B, C, D
The following table gives the descriptive statistics of the output per day of the employees in each program group.
|
SUMMARY TABLE |
||||
|
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 |
Table 5 : Summary measures of Worker Output for each Group (A, B, C, D)
The following table gives the results of the ANOVA where the alpha or level of significance was taken to be 0.05.
|
ANOVA TABLE |
||||||
|
Source of Variation |
Sum of Squares (SS) |
Degree of freedom (df) |
Mean Squares (MS) |
Observed F – statistic |
P- value |
F critical Value |
|
Between Groups variation |
8750 |
3 |
2916.667 |
6.140351 |
0.00557 |
3.238872 |
|
Within Groups variation |
7600 |
16 |
475 |
|||
|
Total variation |
16350 |
19 |
Table 6 : Output for the test for Significance of the Regression Model
The ANOVA table shows that the value of the observed statistic is 6.14 and the critical value is 3.2388. This means that the observed statistic is greater than the critical value and so it is suggested that there exists a difference in the output among the four groups A, B, C and D. The p-value was seen to be 0.005 which is less than the level of significance 0.05 and this too supports the rejection of the null hypothesis which asserts that no difference exists. Then looking at the mean output for the four groups, it is seen that group C has an output of 190 which is markedly greater than the output of group A, B and D. Therefore it is suggested that the company make use of program C to increase the daily productivity of all its line workers.
The weekly sales data of a product of a company of size 7 for one week has been provided and it is of interest to establish the relationship of the weekly sales (y) with that of their competitor’s price (x1) and their own expenditure on advertising (x2).
The following table shows the data given on the same.
|
Week |
Price(x1) |
Advertising(x2) |
Sales(y) |
|
1 |
0.33 |
5 |
20 |
|
2 |
0.25 |
2 |
14 |
|
3 |
0.44 |
7 |
22 |
|
4 |
0.4 |
9 |
21 |
|
5 |
0.35 |
4 |
16 |
|
6 |
0.39 |
8 |
19 |
|
7 |
0.29 |
9 |
15 |
Table 7: Data Given
The regression equation obtained by fitting the given data is given as follows:
y = 3.5976 + 41.32 x1 + 0.0132 x2
The following tables shows the regression output for the fitted model:
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 90.0% |
Upper 90.0% |
|
|
Intercept |
3.597615 |
4.052244 |
0.887808 |
0.424805 |
-7.65322 |
14.84845 |
-5.04115 |
12.23638 |
|
Price of Competitor(x1) |
41.32002 |
13.33736 |
3.098065 |
0.036289 |
4.289567 |
78.35048 |
12.88681 |
69.75324 |
|
Advertising Expenditure (x2) |
0.013242 |
0.327592 |
0.040422 |
0.969694 |
-0.8963 |
0.922782 |
-0.68513 |
0.711617 |
Table 8: Regression Output- Estimated Coefficients and Significance
|
Regression Statistics |
|
|
Multiple R |
0.877814 |
|
R Square |
0.770558 |
|
Adjusted R Square |
0.655837 |
|
Standard Error |
1.83741 |
|
Observations |
7 |
Table 9: Regression model fit measures
The significance of the model is given by the ANOVA table as given below which gives the results of the F-test for significance of the model. The p-value was obtained as 0.052 which is less than the significance level 0.1 or 10% level of significance and hence the significance was found to be significant in explaining variation of weekly sales of the company.
|
Source of Variation |
Degree of freedom (df) |
Sum of Squares (SS) |
Mean Squares (MS) |
Observed F – statistic |
P- value |
|
Regression |
2 |
45.35284 |
22.67642 |
6.716801 |
0.052644 |
|
Residual |
4 |
13.5043 |
3.376075 |
||
|
Total variation |
6 |
58.85714 |
Table 10: Significance test of the regression model
From table the significance of the variable Advertising expenditure of the company was found to be insignificant at 0.1 level as its p-value is 0.969 and thus greater than 0.1. Comparatively the competitor’s price had a p-value of 0.03 which is less than 0.1 and hence significantly related to the weekly product sales of the company.
Based on part (C), the insignificant variable advertising expense was dropped from the regression model and the following regression equation as apparent from the table below was obtained: y =3.5817 + 41.603 x1
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 90% |
Upper 90% |
Lower 90.0% |
Upper 90.0% |
|
|
Intercept |
3.581788 |
3.608215 |
0.992676 |
0.366447 |
-5.69342 |
12.857 |
-3.68894 |
10.85252 |
|
Price of Competitor (x1) |
41.60305 |
10.15521 |
4.096719 |
0.009385 |
15.49825 |
67.70786 |
21.13981 |
62.0663 |
Table 11: Regression Output- Estimated Coefficients and Significance
|
SUMMARY OUTPUT |
|
|
Regression Statistics |
|
|
Multiple R |
0.877761 |
|
R Square |
0.770464 |
|
Adjusted R Square |
0.724557 |
|
Standard Error |
1.643765 |
|
Observations |
7 |
Table 12: Regression model fit measures
|
ANOVA TABLE |
|||||
|
Source of Variation |
Degree of freedom (df) |
Sum of Squares (SS) |
Mean Squares (MS) |
Observed F – statistic |
P- value |
|
Regression |
1 |
45.34733 |
45.34733 |
16.78311 |
0.009385 |
|
Residual |
5 |
13.50981 |
2.701963 |
||
|
Total variation |
6 |
58.85714 |
Table 14: Significance test of the regression model
The new regression model implies that with unit increase in the price of the product by the competitor company, the sales per week of the company increases by 41.603 units and that if the competitor company had been giving away their products for free the weekly sales of the company would be 3.581 units.
