Solution Code: 1HHG
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You are asked to advise a client who is looking into purchasing a cleaning company. It has two Sydney companies in mind (codenamed A and B). The Excel Spreadsheet Prospectives contains the data with the following two variables
Crews: the number of crews sent to each job.
Rooms: the number of rooms cleaned at each job.
As part of its investigation your client is asking you to analyse the data on the two companies which it has obtained and to prepare a report into the following points.
Question 1.Descriptive analysis
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
Iii Give a final summary of your findings from this regression.
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Question 1
The descriptive statistics is given below
Company A | Company B | |||
Number Of Crews | Rooms Clean | Number Of Crews | Rooms Clean | |
Mean | 8.88 | 34.44 | 8.81 | 38.56 |
Standard Error | 0.68 | 2.73 | 0.66 | 2.63 |
Median | 9 | 36 | 8 | 41 |
Mode | 16 | 6 | 16 | 41 |
Standard Deviation | 4.83 | 19.34 | 4.76 | 18.97 |
Sample Variance | 23.37 | 373.88 | 22.63 | 359.94 |
Kurtosis | -1.17 | -0.72 | -1.11 | -0.61 |
Skewness | 0.08 | 0.28 | 0.12 | 0.30 |
Range | 14 | 72 | 14 | 72 |
Minimum | 2 | 6 | 2 | 10 |
Maximum | 16 | 78 | 16 | 82 |
Sum | 444 | 1722 | 458 | 2005 |
Count | 50 | 50 | 52 | 52 |
The mean number of crews sent to each job in company A is 8.88 crews with a standard deviation of 4.83 crews. The median number of crews sent to each job in company A is 9. This indicates that, nearly 50% of the sample number of crews sent to each job falls below 9 and 50% of the sample number of crews sent to each job falls above 9. The minimum and maximum recorded number of crews sent to each job in Company A is 2 and 16 respectively.
The mean number of rooms cleaned to each job in company A is 34.44 crews with a standard deviation of 19.34 crews. The median number of rooms cleaned to each job in company A is 36. This indicates that, nearly 50% of the sample number of rooms cleaned to each job falls below 36 and 50% of the sample number of rooms cleaned to each job falls above 36. The minimum and maximum recorded number of rooms cleaned to each job in Company A is 6 and 78 respectively.
The mean number of crews sent to each job in company B is 8.81 crews with a standard deviation of 4.76 crews. The median number of crews sent to each job in company B is 8. This indicates that, nearly 50% of the sample number of crews sent to each job falls below 8 and 50% of the sample number of crews sent to each job falls above 8. The minimum and maximum recorded number of crews sent to each job in Company B is 2 and 16 respectively.
The mean number of rooms cleaned to each job in company B is 38.56 crews with a standard deviation of 18.97 crews. The median number of rooms cleaned to each job in company B is 44. This indicates that, nearly 50% of the sample number of rooms cleaned to each job falls below 41 and 50% of the sample number of rooms cleaned to each job falls above 41. The minimum and maximum recorded number of rooms cleaned to each job in Company B is 10 and 82 respectively.
Histogram for Number of rooms cleaned – Company A
Going through the histogram of the number of rooms cleaned at each job in Company A, it is found that the distribution of number of rooms cleaned at each jobs follows normal distribution approximately
Histogram for Number of rooms cleaned – Company B
Going through the histogram of the number of rooms cleaned at each job in Company B, it is found that the distribution of number of rooms cleaned at each jobs follows normal distribution approximately
The mean comparison of number of crews and number of rooms cleaned at each job between two companies is given below
Number of Crews | Number of Rooms Cleaned | |||||
Sample Size | Average | Standard Deviation | Sample Size | Average | Standard Deviation | |
Company A | 50 | 8.88 | 4.83 | 50 | 34.44 | 19.34 |
Company B | 52 | 8.81 | 4.76 | 52 | 38.56 | 18.97 |
Grand Total | 102 | 8.84 | 4.77 | 102 | 36.54 | 19.17 |
On comparing the mean number of crews sent to each job between Company A and Company B, we see that Company B sends relatively lesser number of crews when compared to that of Company A
On comparing the mean number of rooms cleaned to each job between Company A and Company B, we see that Company B crews cleans relatively higher number of crews when compared to that of Company A
Question 2
In order to determine whether there is a significant mean difference in OC between Company A and company B, we perform independent sample t test. The null and alternate hypotheses are given below
Null Hypothesis: H0: µOCA = µOCB
That is, the mean OC do not differ significantly between company A and company B
Alternate Hypothesis: H0: µOCA ? µOCB
That is, the mean OC differ significantly between company A and company B
Level of Significance: Let the level of significance be ? = 0.05
Test Statistic
The t test statistic is
t=x1-x2s1n1+1n2
The table given below shows the workings of t test statistic
t-Test: Two-Sample Assuming Equal Variances | ||
OCA | OCB | |
Mean | 3.953083 | 4.734936 |
Variance | 0.883688 | 1.480396 |
Observations | 50 | 52 |
Pooled Variance | 1.188009 | |
Hypothesized Mean Difference | 0 | |
df | 100 | |
t Stat | -3.62161 | |
P(T<=t) one-tail | 0.000231 | |
t Critical one-tail | 1.660234 | |
P(T<=t) two-tail | 0.000462 | |
t Critical two-tail | 1.983971 |
The value of t test statistic is -3.622 and its corresponding p – value is 0.000462. Since the p – value of t test statistic falls below 0.05, there is sufficient evidence to reject the null hypothesis at 5% level of significance. Therefore, we conclude that the mean OC differ significantly between company A and company B. Going through the mean values, we see that the mean OCB (mean = 4.735, Std dev = 1.217) is significantly high when compared to that of OCA (mean = 3.953, Std dev = 0.94), indicating that the company A has the lower mean OC. Therefore, we can recommend Company A to the client
Question 3
The regression model to predict number of rooms cleaned using number of crews as independent variable is given below
The regression model is
Number of Rooms = b0 + b1 * Number of Crews
The regression output is given below
SUMMARY OUTPUT | |||||
Regression Statistics | |||||
Multiple R | 0.925346031 | ||||
R Square | 0.856265276 | ||||
Adjusted R Square | 0.853270803 | ||||
Standard Error | 7.406734177 | ||||
Observations | 50 | ||||
ANOVA | |||||
df | SS | MS | F | Significance F | |
Regression | 1 | 15687.05386 | 15687.05386 | 285.9485 | 7.46E-22 |
Residual | 48 | 2633.266136 | 54.85971116 | ||
Total | 49 | 18320.32 | |||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | |
Intercept | 1.575440067 | 2.207798679 | 0.713579586 | 0.478943 | -2.86364 |
Number Of Crews | 3.700963956 | 0.21886228 | 16.91001277 | 7.46E-22 | 3.260912 |
The regression equation is
Number of Rooms = 1.575 + 3.701 * Number of Crews
The independent variable is number of crews and the dependent variable is number of rooms cleaned. Here, we wishes to predict the number of rooms cleaned using the number of crews employed at each job.
iii.
The value of f test statistic is 285.9485 and its corresponding p – value falls well below 0.05, indicating that the regression model is good fit in predicting the number of rooms cleaned at each job. In addition, we see that there exists a strong positive linear relationship between number of crews and number of rooms cleaned (Correlation Coefficient = 0.9253). Thus, we can conclude that the independent variable number of crews at each job is a significant predictor of number of rooms cleaned
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