BUS201 Foundations of Business Success | Business Management
Questions:
Question 1.
Descriptive analysis
- Prepare appropriate summary statistics on each variable and comment on each of these.
- Use Excel to draw up the histograms of number of rooms cleaned for each company. Comment on the shape of the histograms and compare between the two companies.
- Use a PivotTable to summarise the performance of the number of crews by looking at the sum of rooms cleaned by each crew size for each company. Compare the two companies by using the output of the pivot table.
Question 2.
Create a new variable for each company, viz., the ratio of The number of Offices cleaned
Number of crews
Call these variables OCA and OCB
Your client wants to know which company has the lower mean OC (why does he want this statistic?). Other information suggests that company B might have the higher mean. Test this hypothesis at 5% level of significance. Set out your analysis for this fully. What can be recommended to your client?
Question 3.
For the company chosen in 2, your boss wants you to fit a simple linear regression model to the
data. Incorporate appropriate output in your report.
y = Number of Rooms cleaned
x = Number of Crews
- Justify the choice of explanatory and dependent variables.
- Estimate the regression equation between these variables.
- State and interpret the slope coefficient ii.
State and interpret the R2 value from this.
Iii Give a final summary of your findings from this regression.
Answers:
Solution to Question 1
We have 2 variables each for 2 firms A and B, which makes a total of 4 variables. The descriptive statistics are shown in the table:
|
|
A |
B | ||
|
|
Crews |
Rooms |
Crews |
Rooms |
|
Mean |
8.880 |
34.440 |
8.808 |
38.558 |
|
Standard Error |
0.684 |
2.735 |
0.660 |
2.631 |
|
Median |
9.000 |
36.000 |
8.000 |
41.000 |
|
Mode |
16.000 |
6.000 |
16.000 |
41.000 |
|
Standard Deviation |
4.835 |
19.336 |
4.757 |
18.972 |
|
Sample Variance |
23.373 |
373.884 |
22.629 |
359.938 |
|
Kurtosis |
-1.167 |
-0.720 |
-1.109 |
-0.608 |
|
Skewness |
0.081 |
0.279 |
0.124 |
0.295 |
|
Range |
14.000 |
72.000 |
14.000 |
72.000 |
|
Minimum |
2.000 |
6.000 |
2.000 |
10.000 |
|
Maximum |
16.000 |
78.000 |
16.000 |
82.000 |
- Let us look at crew size first. The average size is almost equal for A and B - 8.88 for A and 8.808 for B.
- The no of rooms cleaned by the crew is different across A and B. The average number of rooms cleaned equals 34.4 rooms/crew for A and 38.5 rooms/crew for B. This shows that crew in firm B are more efficient as each crew member cleans more rooms than a member in A.
- In terms of distribution all variables are positively skewed. The skewness for rooms cleaned is almost same for A and B: 0.279 and 0.295. The skewness for crew size is greater for B at 0.124 against only 0.081 for A.
- The extent of data spread is measured with variance or its square root- standard deviation. Both these are absolute measures that depend on units. The spread is almost same for both firms and variables. These values are 4.8 and 4.7 for no of crew A and B respectively. The figures for variance of rooms cleaned is 19.3 for A and 18.9 for B.
- We can choose a relative measure of spread called coefficient of variation(FAQs). It is calculated as the ratio of standard deviation to mean value. This value is same for crew size across firms at 0.54, but is higher for rooms cleaned by A at 0.56 as compared to 0.49 for B.
|
CV |
0.544433 |
0.561443 |
0.540094999 |
0.492042578 |
Next we do a visual presentation of data using histograms as shown below. Note that the class intervals , class width and no of classes depend on the data with us. The data values for crew size is similar in A and B. There is similarity in the maximum and minimum values, which leads to almost equal range value. This allows us to choose identical classes for this variable for A and B. The high degree of similarity is visible as both have zero entries for the crew size 12-15.
The story is not same for rooms cleaned. The maximum and minimum values are different, leading to intervals with different width, limits and number.
The pivot table shows the following results: B has more crews ( 458>444) and it also does more rooms.( 2005 > 1722)
|
DATA for SUM |
|
|
Sum of Number Of Crews A |
444 |
|
Sum of Rooms Clean A |
1722 |
|
Sum of Number Of Crews B |
458 |
|
Sum of Rooms Clean B |
2005 |
Answer 2
The OC is shown below:
|
OC A |
OC B |
| ||
|
2.875 |
7.5 |
| ||
|
3.75 |
7 |
| ||
|
3.75 |
4.25 |
| ||
|
4.5 |
9 |
| ||
|
3.92 |
5.167 |
| ||
|
3.92 |
6.5 |
| ||
|
4.125 |
4.25 |
| ||
|
3.08 |
5.5 |
| ||
|
2.81 |
3.833 |
| ||
|
4.58 |
5 |
| ||
|
3.167 |
4.6 |
| ||
|
4.0625 |
5.7 |
| ||
|
4.1 |
4.312 |
| ||
|
4.75 |
6.5 |
| ||
|
4.5 |
4.5 |
| ||
|
2.875 |
4.917 |
| ||
|
3.7 |
5.125 |
| ||
|
5.2 |
3.417 |
| ||
|
2.25 |
3.25 |
| ||
|
2.75 |
5 |
| ||
|
4.875 |
4 |
| ||
|
4.33 |
6.25 |
| ||
|
4.625 |
4.1 |
| ||
|
3.5 |
3.75 |
| ||
|
3 |
4.437 |
| ||
|
4.75 |
5 |
| ||
|
7 |
3.667 |
| ||
|
2.875 |
4.625 |
| ||
|
4.2 |
3.417 |
| ||
|
3.187 |
4.2 |
| ||
|
3.25 |
4.375 |
| ||
|
2.75 |
5 |
| ||
|
4.583 |
3 |
| ||
|
4.3125 |
4.75 |
| ||
|
4.0625 |
3.125 |
| ||
|
4.125 |
5.6 |
| ||
|
3.3 |
5 |
| ||
|
5 |
4.625 |
| ||
|
5.5 |
5.5 |
| ||
|
5.5 |
5.125 |
| ||
|
4.5 |
3.833 |
| ||
|
5 |
5.25 |
| ||
|
3 |
3.25 |
| ||
|
3.8 |
4.1 |
| ||
|
3.875 |
7 |
| ||
|
5.5 |
3.625 |
| ||
|
3 |
4.917 |
| ||
|
3.417 |
3.5 |
| ||
|
3.17 |
4.5625 |
| ||
|
3 |
4.75 |
| ||
|
|
3.1875 |
| ||
|
|
3.375 |
| ||
|
|
OC A |
OC B | ||
|
Mean |
3.95 |
4.73 | ||
|
Standard Error |
0.13 |
0.17 | ||
|
Median |
3.92 |
4.61 | ||
|
Mode |
3 |
5 | ||
|
Standard Deviation |
0.94 |
1.22 | ||
|
Sample Variance |
0.88 |
1.48 | ||
|
Kurtosis |
0.78 |
2.03 | ||
|
Skewness |
0.68 |
1.18 | ||
|
Range |
4.75 |
6 | ||
|
Minimum |
2.25 |
3 | ||
|
Maximum |
7 |
9 | ||
|
Sum |
197.65 |
246.22 | ||
|
Count |
50 |
52 | ||
- The observations for A and B are unequal - A has 50, while B has 52.
- The average OC higher for B at 4.73, as compared to 3.95 for.
- The other measures of central tendency- median and mode are higher for B.
- The OC values are positively skewed, and skewness is lower for A (1.18 > 0.68).
- The extent of peakedness is greater for B (2.03) as compared to A (0.78).
Statistical Test For Difference In Oc
We use a 1 tail test as we have to see if B has a higher average OC than a(5 steps to hypothesis test)
We can use a z test for difference in means, as the sample size exceeds 30 for both data series.
Step 1: Null hypothesis setting
Ho: µA ≤ µB
Step 2: Alternative hypothesis setting
Ha: µA ≤ µB
Step 3: Choice of the significance value, α
Let α = 0.05 so that the corresponding z critical value is -1.645, ( minus shows rejection region is on left side).
Step 4: critical z value
The test value equals -3.643
Step 5: comparison of test and critical value.
In this case, test value > critical value ignoring the minus sign. So DO NOT ACCEPT the null hypothesis. (The fice steps for hypothesis testing )
Step 6: Conclusion
There is statistical support at 95% confidence level that B has a higher OC than A.
This leads us to recommend B.
Answer 3
The regression analysis is done for B. (Regression analysis)
Regression line is Y= α +ßX
Y = Number of Rooms cleaned
X = Number of Crews
α is intercept
ß is slope .
The choice of explanatory and dependent variable assumes that it is the availability of crews that decides how many rooms can be cleaned.
- The estimated equation is Y= 0.912 +0.231*X
- The regression coefficient is positive, so the variables are positively related.
- The equation is linear in variables, ruling out any non linear relation between them.
- The slope = 0.231, and it represents the additional value of Y. For 1 more crew we can clean 0.231 rooms more on an average.
- R2= 0.856. This shows a good fit as 85.6% of variation in Y is explained by variation in X value.
- Improvements:
Add new explanatory variables
Try a non linear model like quadratic or polynomial.
References
5 steps to hypothesis test . (n.d.). Retrieved june 5, 2017, from Online courses.science.psu.edu.
FAQs. (n.d.). Retrieved june 6, 2017, from Stats.idre.ucla.edu: https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faq-what-is-the-coefficient-of-variation/
Regression analysis. (n.d.). Retrieved June 6, 2017, from Home.iitk.ac.in: https://home.iitk.ac.in/~shalab/regression/Chapter2-Regression-SimpleLinearRegressionAnalysis.pdf
The five steps for hypothesis testing . (n.d.). Retrieved June 5, 2017, from LEarn.bu.edu: https://learn.bu.edu/bbcswebdav/pid-826908-dt-content-rid-2073693_1/courses/13sprgmetcj702_ol/week04/metcj702_W04S01T05_fivesteps.html
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