HI6007 Group Assignment

Construct a frequency distribution, cumulative frequency distribution, relative frequency distribution, cumulative relative frequency distribution and percent frequency distribution for the data set using a class width of 10. (5 marks)

 

Construct a histogram showing the percent frequency distribution of the examination scores. Comment on the shape of the distribution.

Below you are given the examination scores of 20 students (data set also provided in accompanying MS Excel file).

Answer

Class	Frequency	Cumulative frequency	Relative frequency	Cumulative relative frequency	Percent frequency
50-59	3	3	0.15	0.15	15%
60-69	2	5	0.1	0.25	25%
70-79	5	10	0.25	0.5	50%
80-89	4	14	0.2	0.7	70%
90-99	6	20	0.3	1	100%

1. Construct a histogram showing the percent frequency distribution of the examination Comment on the shape of the distribution. (2 marks)

The shape of the histogram shows a kind of left skewed distribution, this shows that the distribution is negatively skewed. However, the skewness is not severe.

Question 2 (8 marks)

Shown below is a portion of a computer output for a regression analysis relating supply (Y in thousands of units) and unit price (X in thousands of dollars)

1. What has been the sample size for this problem? (1 mark)

Sample size = 39+1+1 = 41

1. Determine whether or not supply and unit price are related. Use α = 0.05. (2 marks)

We obtain the t-value

The t-critical value is 2.022; this value is greater than the computed value hence the null hypothesis is not rejected. This means that supply and unit price are not related at α = 0.05.

1. Compute the coefficient of determination and fully interpret its meaning. Be very specific. (2 marks)

The coefficient of determination is 0.048; this implies that unit price only explains 4.8% of the variation in the dependent variable

1. Compute the coefficient of correlation and explain the relationship between supply and unit price. (2 marks)

The coefficient of correlation is 0.21921; this indicates that there is a weak positive relationship between supply and unit price.

1. Predict the supply (in units) when the unit price is $50,000. (1 mark)

The regression equation model is;

Thus the supply is approximately 1505.

Question 3 (6 marks)

Allied Corporation wants to increase the productivity of its line workers. Four different programs have been suggested to help increase productivity. Twenty employees, making up a sample, have been randomly assigned to one of the four programs and their output for a day's work has been recorded. You are given the results below (data set also provided in accompanying MS Excel file).

1. Construct an ANOVA table. (3 marks)

Answer

The following is the ANOVA Table

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	2916.667	6.140351	0.00557	3.238872
Within Groups	7600	16	475
Total	16350	19

1. As the statistical consultant to Allied, what would you advise them? Use a .05 level of significance. (3 marks)

Answer

Looking at the above results, it can be seen that the p-value is 0.00557 (a value less than α = 0.05). With this, we therefore reject the null hypothesis and conclude that the productivity varies based on the program. Results further showed that Program D had significantly higher productivity than any other program. As the statistical consultant to Allied, I would advise them to consider program D since more productivity would be realized from this program.

Question 4 (9 marks)

A company has recorded data on the weekly sales for its product (y), the unit price of the competitor's product (x1), and advertising expenditures (x2). The data resulting from a random sample of 7 weeks follows. Use Excel's Regression Tool to answer the following questions (data set also provided in accompanying MS Excel file).

1. What is the estimated regression equation? Show the regression output. (2 marks)

Answer

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.877814
R Square	0.770558
Adjusted R Square	0.655837
Standard Error	1.83741
Observations	7

 

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.597615	4.052244	0.887808	0.424805	-7.65322	14.84845
Price	41.32002	13.33736	3.098065	0.036289	4.289567	78.35048
Advertising	0.013242	0.327592	0.040422	0.969694	-0.8963	0.922782

The estimated regression equation is given below;

1. Determine whether the model is significant overall. Use α = 0.10. (2 marks)

Answer

ANOVA
	df	SS	MS	F	Significance F
Regression	2	45.35284	22.67642	6.716801	0.052644
Residual	4	13.5043	3.376075
Total	6	58.85714

As can be seen I the above table, the p-value for the F-statistics is 0.0526 (a value less than α = 0.10), this leads to rejection of the null hypothesis and thus we conclude that the overall model is significant at α = 0.10.

1. Determine if competitor’s price and advertising is individually significantly related to sales. Use α = 0.10. (2 marks)

Answer

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.597615	4.052244	0.887808	0.424805	-7.65322	14.84845
Price	41.32002	13.33736	3.098065	0.036289	4.289567	78.35048
Advertising	0.013242	0.327592	0.040422	0.969694	-0.8963	0.922782

The p-value for the price is 0.036 (a value less than α = 0.10), this leads to rejection of the null hypothesis and thus we conclude that competitor’s price is individually significantly related to sales at α = 0.10.

The p-value for the advertising is 0.9697 (a value greater than α = 0.10), this leads to non-rejection of the null hypothesis and thus we conclude that advertising is individually not significantly related to sales at α = 0.10.

1. Based on your answer to part (c), drop any insignificant independent variable(s) and re-estimate the model. What is the new estimated regression equation? (2 marks)

Answer

We drop the advertising and the new results as shown below;

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.877761
R Square	0.770464
Adjusted R Square	0.724557
Standard Error	1.643765
Observations	7

 

ANOVA
	df	SS	MS	F	Significance F
Regression	1	45.34733	45.34733	16.78311	0.009385
Residual	5	13.50981	2.701963
Total	6	58.85714

 

	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	3.581788	3.608215	0.992676	0.366447	-5.69342
Price	41.60305	10.15521	4.096719	0.009385	15.49825

The estimated regression equation is given below

1. Interpret the slope coefficient(s) of the model from part (d). (1 marks)

Answer

The slope coefficient for the competitor’s price is 41.6031; this implies that a unit increase in the competitor’s price would result to an increase in the sales made by 41.6031. Similarly, a unit decrease in the competitor’s price would result to a decrease in the sales made by 41.6031.