Final exam teachers : François Bellavance and Jean-Claude Lebrun

FINAL EXAM

Teachers : François Bellavance and Jean-Claude Lebrun

Note : 50 people (18 francophones and 32 anglophones or others) did not know or did not answer the question and thus were not entered in this table.

1. 
   In Montreal, is there a significant difference in the level of confidence granted to the mayor between the francophones and the anglophones (or others)? Use a =5% level and verify, at least 3 TIMES, that there is no errors in the transcription of the data in EXCEL or MINITAB,!!!! (2 points)
   

2. 
   If you observed a significant link, briefly describe this link. If you did not observe a significant link, briefly describe why. (4 points)
   

During the same survey, for the question, « If a referendum was held on the island of Montreal, would you vote yes or no to the following question : do you want to replace the 29 municipalities of the island of Montreal by one city, i.e. one island, one city? », we obtained the following results :

Note : 33 people (18 francophones and 15 anglophones or others) did not know or did not answer the question and thus were not entered in this table.

We want to verify the hypothesis that in Montreal, the proportion of francophones in favour of the « one island, one city» project is different from the proportion of anglophones (or others).

3. 
   Formulate the hypotheses for this problem.(1 point)
   

4. 
   In the sample, what are the respective proportions of francophones and anglophones in favour of the « one island, one city» project? (1 point)
   

5. 
   Obtain the « p-value » for the test of the hypotheses formulated in c) and give your conclusion at the  =5 % level? (4 points)
   

A firm of« head hunters » offers its services to recruit the best managers either from inside or outside the company. In the business world, several people claim that the managers hired from outside of the company obtain better performances than the ones recruited from within the company.

1. 
   Using EXCEL (or MINITAB) obtain the minimum, the maximum, the mean and the standard deviation of the 150 managers salaries for the sample? (4 points)
   

Minimum :

2. 
   Using EXCEL (or MINITAB), obtain the 95% confidence interval for the mean of the United States managers salaries and briefly give the interpretation of that interval. (4 points)
   

3. 
   Starting from the confidence interval for the mean of the salaries found in b), define if the « p-value » to confront the hypotheses H₀ :  = 70 against H₁ :   70 , where  represents the true mean of the United States managers salaries in thousands of U.S. dollars, would be higher, lower or equal to 5%. Briefly justify your answer. (2 points)
   

1. 
   Formulate precisely the hypotheses H₀ and H₁ that we want to confront in this problem. (1 point)
   

2. 
   In the sample, what is the proportion of managers recruited from outside the company?(2 points)
   

3. 
   Using EXCEL (or MINITAB), obtain the p-value corresponding to your hypotheses formulated in a) and give your conclusion at the =5% level? (3 points)
   

4. 
   Using EXCEL (or MINITAB), obtain the 95% confidence interval for the proportion of the United States managers recruited from outside the company and briefly give the interpretation of this interval. (4 points)
   

5. 
   Would you have been able to verify the hypotheses formulated in a) if instead of having taken a simple random sample, the researchers’ team had used a stratified sampling design with the managers recruited externally as first strata and the managers recruited within the company as second strata? Briefly justify your answer. (5 points)
   

1. 
   What are the means and the standard deviations of the score of performance for the groups of managers recruited internally and externally respectively? (4 points)
   

External  mean of performance: standard deviation :

2. 
   We are now interested to verify the hypothesis that on average the managers recruited from outside the company obtain higher scores of performance than the ones recruited from within the company. Previously we carried out a test on the variances in order to take the good statistical test to compare the means. We obtained the following results for the test on the variances:
   

H₀ : equal variances H₁ : unequal variances p-value = 0.309.

Salary Years
Origin Years Outside
Origin Origin

Correlations (Pearson)

P-Value
Performance Salary

Salary 0.684

0.000
Years 0.068 -0.323

0.410 0.000

In order to greater analyse and understand this set of data and the relations between the variables, it is also interesting to examine the scatterplots between the performance, the salary and the years of experience by identifying on scatterplots the two manager groups. The Pearson correlation coefficients between these variables were also calculated separately for each of the two groups.

Salary Years O external + internal O external + internal
Years O external + internal

Managers recruited from outside the company (external) Correlations (Pearson) P-Value Performance Salary Salary 0.736 0.000 Years 0.150 -0.174 0.245 0.175

Managers recruited within the company (internal) Correlations (Pearson) P-Value Performance Salary Salary 0.642 0.000 Years 0.276 -0.014 0.009 0.899

Questions :

1. 
   According to the graphs and the Pearson correlation coefficients, we note that for all the managers there is a negative linear relation which is significant (r = -0.323) between the salary and the number of years of experience. How would you explain this relation which, at first sight, seems to be somewhat unexpected? (5 points)
   

Using SAS software, we have obtained a summary of all the multiple linear regression models characteristics. The results are the following :

Model R-square

1 0.05674645 0.05037311 167.17715 EXT-INT

1 0.00458214 -.00214366 184.49664 YEARS

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2 0.56021507 0.55423160 2.01651 SALARY YEARS

2 0.48849467 0.48153541 25.82897 SALARY EXT-INT

2 0.11058043 0.09847948 151.30330 YEARS EXT-INT

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3 0.56026479 0.55122914 4.00000 SALARY YEARS EXT-INT

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2. 
   Which one of the various multiple and simple linear regression models seems to be the best and why ? (4 points)
   

2. 
   Using EXCEL (or MINITAB), obtain the linear regression line for the best model found in b) and briefly interpret the coefficients of this model as well as the squared coefficient of determination. (6 points)
   

1. 
   2 by 3 crossed table. Yes there is a significant difference since the p-value = 0.000007 < 0.05. So, we reject the hypothesis H₀ : there is no link between the confidence level granted to the mayor by the Montreal francophones comparatively to the anglophones (or others). (The coefficient of Cramer = 0.2288)
   

2. 
   The proportion of Montrealers who have less confidence in the mayor is similar for francophones and anglophones (38.56% and 39.60% respectively). However, francophones have more confidence in the mayor in a proportion of only 10.46% comparatively to 26.85% for anglophones. On the other hand, the proportion of Montrealears who have the same level of confidence in the mayor is 50.98% for francophones and 33.56% for anglophones.
   

3. 
   H₀ : p_francophones = p_anglophones vs H₁ : p_francophones  p_anglophones .
   

5. 
   p-value = 0.0035 <  = 0.05. Consequently, we reject the hypothesis H₀ . So, in Montreal, the proportion of francophones in favour of the « one island, one city» project is significantly different from the proportion of anglophones (or others) in favour of the project.
   

1. 
   Minimum = 48, Maximum = 103, Mean = 71.63 and standard deviation = 10.704
   

2. 
   95% CI (69.906 ; 73.360). By saying that the true mean of the United States managers salaries is between 69,906$ and 73,360$, there is only 5% chance of error.
   

3. 
   The p-value to confront these hypotheses would be > 0.05 because 70,000 is included in the 95% confidence interval .
   

1. 
   H₀ : p_external  50% against H₁ : p_external < 50%.
   

3. 
   p-value = 0.0169 <  = 0.05. Consequently, we reject the hypothesis H₀ . So, the proportion of managers recruited from outside the company is significantly lower than 50%.
   

4. 
   95% CI (33.45% ; 49.21 By saying that the true proportion of managers recruited from outside the company is between 33.45% and 49.21% , there is only 5% chance of error.
   

5. 
   No. In the case of a stratified sampling where the two strata are the managers recruited from within
   

1. 
   External  performance mean : 6.32 standard deviation : 1.342
   

2. 
   H₀ : _external  _internal vs H₁ : _external > _internal . Test to compare two means with equal variances  (because we do not reject the equality of variances, p-value = 0.309 > 0.05): p-value = 0.001665 <  = 0.05. Consequently, we reject the hypothesis H₀ . Thus, the mean of the managers scores of performance hired from outside the company is significantly higher than the mean of the managers scores of performance recruited from within the company.
   

1. 
   By Analysing the scatter plots, we observe that the « externals » have on average a higher salary than the « internals » while having on average less years of experience. Also when we look separately at the relation between the salaries and the years of experience for the « externals » and the «internals», the link is no longer significant (externals : r = -0.174 p-value = 0.175 ; iternals : r = -0.014 p-value = 0.899).
   

2. 
   The model with salaries and years of experience. Comparatively to the other models, this model has the greatest value of R² _ajusted (55.4%) and the smallest value of C_p (2.01).
   

3. 
   Performance = -2.9206 + 0.1093 x salary + 0.1215 x years of experience. R² = 56.02%, therefore, 56.02% of the observed variability in the managers scores of performance is explained by the salaries and the years of experience. According to the model, when the salary is higher, the score of performance is higher. Also, when the number of years of experience is higher, the score of performance is higher.