1. Draw a random sample of two hundred and fifty (250) households as per the sample selection procedure. What sampling method have you used to select your sample data? In your opinion, is this the best method of sampling particularly when one is interested in characteristics like the gender of the household head, education levels etc., why or why not?

2. Compute the descriptive statistics and draw a Box-Whisker plot of Expenditures on the following variables (all series in one graph!);

(i) Alcohol (ii) Meals (iii) Fuel (iv) Phone

3. Use information from the descriptive statistics and the boxplots in part (B) above to present a summary of your findings by contrasting different features of these distributions.

4. Find the top 5% value and the bottom 5% value of household’s annual after-tax income (AtaxInc). What do these two values imply?

5. The series OwnHouse represents whether a household owns a house or not. Let X be a random variable such that X = Number of households who own a house.

6. Construct a contingency table between the gender and the level of education.

7. What is the probability that the head of household is a male and his highest level of education is Intermediate?

8. What is the probability that the head of household is a female and has the Bachelor degree?

9. What is the proportion of having the Secondary as the highest degree from among males?

10. Do you think that the events "gender of household head is female" and "having the Master Degree" are independent?

## Measures of Central Tendency, Skewness, and Distribution Shape

The requisite sampling method that has been used is simple random sampling. In this sampling method, there is an equal probability associated with the selection of each element. A key concern in this sampling method is the under-representation and over-representation of key population attributes especially taking into cognizance the low sample size (Eriksson & Kovalainen, 2015). Thus, it would be better to sue stratified random sampling since it would ensure that there is fair representation of the important population attributes. This method involves classification of population as per key attributes as the first step which is followed by random sampling of requisite individuals from each group who are selected in the same ratio as they are present in the population.(Flick, 2015).

Descriptive statistics along with the Box- whisker plot for the variables (Alcohol, meals, fuel and phone) is given below:

Based on the descriptive statistics, it can be seen that for all the four variables the measures of central tendency i.e. mean median and modes are not same. This indicates that the distribution of data is not normal and thus shape can be assumed to be asymmetric. Further, this is also evident from the non-zero values of skew for each of the given variables. This is because value of skew must be zero for normal distribution of data. Moreover, the high positive value of skew also represents the presence of outliers at the high positive side of data which tends to have a distorting effect on mean, thus making the median a more favourable choice (Hastie, Tibshirani & Friedman, 2011).

Task 2

- For the variable, utilities the frequency distribution table is highlighted below:

- The requisite percentage of households who spend on utilities is computed as highlighted below:

a. “At the most $900 per annum” Number of households = 250 Total number of households which have spent at most $900 per annum on the variable utilities = 94 Hence, Percentage of households which have spent at most $900 per annum on the variable utilities = Total number of households which have spent at most $900 per annum on the variable utilities / Number of households Therefore, the percentage of households which have spent at most $900 per annum on the variable utilities is 37.6%. |

b. “Between $1500 and $2700 per annum” Number of households = 250 Total number of households which have spent between $1500 and $2700 per annum on the variable utilities = 83+49 = 132 Hence, Percentage of households which have spent between $1500 and $2700 per annum on the variable utilities = Total number of households which have spent between $1500 and $2700 per annum on the variable utilities / Number of households Therefore, the percentage of households has spent between $1500 and $2700 per annum on the variable utilities 52.80%. |

c. “More than $3000 per annum” Number of households = 250 Total number of households which have spent more than $3000 per annum on the variable utilities = 3 Hence, Percentage of households which have spent more than $3000 per annum on the variable utilities = Total number of households which have spent more than $3000 per annum on the variable utilities / Number of households Therefore, the percentage of households which have spent at most $900 per annum on the variable utilities is 1.20%. |

The top 5% value of household’s for variable annual tax income (Ataxlnc) and bottom 5% value of household’s for variable annual tax income (Ataxlnc).

Based on the above two values, it can be said that the top 5% value is the representation of the fact that 95% of households will have the after tax income (Ataxlnc) lesser than $143,023.30. Similarly, the bottom 5% value is the representation of the fact that 95% of households will have the after tax income (Ataxlnc) greater than $50,291.50 (Hair et. al., 2015).

(i) It is apparent from the data sheet that variable OwnHouse has a numerical value of either 0 or 1 that represents that the house is either owned or rented. Hence, the variable x would be considered a quantitative variable (Hastie, Tibshirani & Friedman, 2011).

(ii) It can be said that when only one household is taken into consideration, then in such cases the possible events for variable x would be only 2 i.e. either 0 or 1. Hence, the probability distribution would be assumed to be normally distributed (Hair et. al., 2015). Whereas, when 250 households would be taken into account, then the probability distribution would be Poisson distribution because X would have discrete integral values. Therefore, the probability distribution cannot be assumed to be continuous normal distribution for the X value when 250 households are considered (Flick, 2015).

## Analysis of Expenditure Levels

Scatter plot for the natural log of variables i.e. after tax income and total expenditure is highlighted below:

Independent variable: After tax income

Dependent variable: Total expenditure

From the value of correlation coefficient and above show scatter plot, the conclusion can be drawn that the strength of correlation between the variables is moderate. The medium strength of correlation is representation of the fact that the household with high level of after tax income would have higher level of total expenditure (Flick, 2015).

In order to represent the correlation between the variables level of education and gender, a contingency table is used which is shown below:

Probability that household head will have Intermediate level of education and is a male.

Total households = 250

Number of male household head with Intermediate level of education = 27

Probability = Number of male household head with Intermediate degree / Total households = 27 / 250 = 0.108

Hence, 10.8% or 0.108 probability that household head will have Intermediate level of education and is a male.

- Probability that household head will have Bachelor level of education and is a female.

Total households = 250

Number of female household head with Bachelor level of education =24

Probability = Number of female household head with Bachelor level of education / Total households = 24 / 250 = 0.096

Hence, 9.6 % or 0.096 is probability that household head will have Bachelor level of education and is a female.

- Proportion of number of households which have male household who holds Secondary level of education.

Total male households head = 130

Number of male household head with Secondary level of education =27

Proportion of number of households which have male household who holds Secondary level of education = Number of male household head with Secondary level of education/ Total male households head = 27/130 = 0.2077

Hence, 0.2077 proportion of number of households which have male household who holds Secondary level of education.

Case X and case Y would be independent only when (Fehr & Grossman, 2013).

It is apparent that the above condition is not satisfied and thus, the cases (having master degree and female household head) are not independent.

References

Eriksson, P. & Kovalainen, A. (2015) Quantitative methods in business research (3^{rd} ed.). London: Sage Publications.

Fehr, F. H., & Grossman, G. (2013) An introduction to sets, probability and hypothesis testing (3^{rd} ed.). Ohio: Heath.

Flick, U. (2015) Introducing research methodology: A beginner's guide to doing a research project (4^{th} ed.). New York: Sage Publications.

Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., & Page, M. J. (2015) Essentials of business research methods (2^{nd} ed.). New York: Routledge.

Hastie, T., Tibshirani, R. & Friedman, J. (2011) The Elements of Statistical Learning (4^{th} ed.). New York: Springer Publications.

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