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STA 101 : Statistics

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STA 101 : Statistics

0 Download3 Pages / 552 Words

Course Code: STA101
University: Vesalius College

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Country: Belgium

Question: 

1. During the recession that began in 2008, not only did some people stop making house payments, but they also stopped making payments for local government services such as trash collection and water and sewer services. The following data have been collected by an accountant who is performing an audit of account balances for a major city billing department. The population from which the data were collected represents those accounts for which the customer had indicated the balance was incorrect. The dependent variable, y, is the actual account balance as verified by the accountant. The independent variable, x, is the computer-generated account balance.

X

280

12

29

67

94

122

108

240

413

144

22

Y

319

16

29

73

117

134

111

247

416

156

30

 
a. Compute the least squares regression equation.
b. If the computer-generated account balance was 100, what would you expect to be the actual account balance as verified by the accountant?
c. The computer-generated balance for Oliver Buxton is listed as 100 in the computer-generated account record. Calculate a 95% interval estimate for Mr. Buxton’s actual account balance.
d. Calculate also a 95% interval estimate for the average of all customers’ actual account balances in which a computer-generated account balance is the same as that of Mr. Jones (part c). Interpret your results.
2. Statistical analysis of has shown a strong negative correlation between skiing accidents and consumption of ice-cream in a data set of various countries in 2017. Are these statistical findings a proof of causal relationship between these two variables? Explain.

Answer: 
1. Table 1: Simple linear regression model
                                           

a). The least square regression equation assuming ‘The actual account balance’ as dependent variable (y) and ‘The computer-generated account balance’ as independent variable (x) is found out to be-
‘The actual account balance’ = 7.183895565 + 1.024805453 * ‘The computer generated balance’ (Neter et al. 1996).
b). If the computer-generate balance is 100, then the expected actual account balance as verified by the accountant-
‘The actual account balance’ = 7.183895565 + 1.024805453 * ‘The computer generated balance’.
Or, ‘The actual account balance’ = 7.183895565 + 1.024805453 * 100
Or, ‘The actual account balance’ = 7.183895565 + 102.4805453
Or, ‘The actual account balance’ = 109.6644409
c). The upper 95% confidence interval of the actual balance is estimated by the regression model-
‘The actual account balance’ = -4.84305113 + 0.959108326 * ‘The computer generated balance’.
‘The actual account balance’ = 19.2108423 + 1.09050258 * ‘The computer generated balance’.
When, the computer generated balance of Mr. Oliver Buxton is accounted as 100, then the upper and lower 95% confidence intervals of actual balances are respectively- 128.2611 and 91.06778.
That means, it is 95% probable that actual account balance of Mr. Oliver Buxton ranges in the interval (91.06778, 128.2611).
d). The computer generated balance remains same. The linear regression equations provide the upper 95% and lower 95% confidence limits of actual account balances of 11 people. The averages of predicted actual balance are-
170.98897 (Upper 95% confidence interval of the average of estimated actual account balance).
128.64739 (Lower 95% confidence interval of the average of estimated actual account balance).
It indicates that there exists 95% probability that the average actual account balance of all the samples lie in the interval of (128.64739 and 170.98897).
Table 2: Table of 95% confidence intervals of average of actual account balance
                                             
Alternate 1.d)
The average account balance = 149.8182.
The standrad account balance = 128.2785
95% confidence interval of account balance = 75.80773
The upper and lower 95% confidence limits of the average of account balance are = [74.01046, 225.6259].
2. A statistical analysis shows that a strong negative correlation exists between ‘consumption of ice-cream’ and ‘skiing accidents’ for the analysis of data set of various countries in 2017. However, theses two fcators do not have any ’cause and effect’ relationship. These two factors are absolutely independent to each other (Box 1966). Therefore, such type of correlation is actually the ‘Non-sense correlation’.
It is the term used to describe a circumstances where two variables are correlated without being causalty associated to each other. Seldom, a third variable could be found out to link a association between two variables. For example, the year wise consumption of food-grain in Europe and life-expecatncy of people of Africa could be highly and positively correlated. Also, the use of solar energy and growth of Malaria in Asian countries could be highly and negatively correlated. However, there exists no significant and explanatory association or ’cause and effcet’ relation between the pairs of variables. Hence, these cause non-sense correlation.   
References:
Box, G.E., 1966. Use and abuse of regression. Technometrics, 8(4), pp.625-629.
Neter, J., Kutner, M.H., Nachtsheim, C.J. and Wasserman, W., 1996. Applied linear statistical models (Vol. 4, p. 318). Chicago: Irwin.

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