Bus5Sbf Statistics For Business: Analysing Assessment Answer

Tasks for Analysis of Data Set

Complete the following tasks based on the unique data set you generated.

Task 1

1. Draw a random sample of two hundred (200) 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, 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

1.  Also, use an appropriate measure of variation tocompare the variability in expenditures on these four variables. Explain, why is this an appropriate measure.
2.  Present a summary of your findings about the shape and spread of the distribution of these variables using information from the boxplots and the descriptive statistics.

Task 2

1.  Construct a frequency distribution of the expenditures on Utilities, using the following classification ;

1.  What is the percentage of households that spend on Utilities
2. at the most $ 1200 per annum
3. between $1200 and $2400 per annum, and
4. more than $2400 per annum.

1.  Draw the histogram of the expenditures on Utilitiesby households in your sample. Do you think the utility expenditures are normally distributed? Provide the “statistical reason” for your answer?

Task 3

1.  What is the top 10% value and the bottom 10 % value of household’s annual after tax income (AtaxInc)? What does these two values imply?
2.  What does the mean (average) of variable OwnHouseimply?
3.  What is the probability that a randomly selected household will have a family size (FS= Adults + Children) equal to 5?
4.  Draw a scatter plot of natural log of total expenditures against natural log of after tax income, that is,  ln(texp) against ln(ataxinc) and compute the coefficient of correlation. Express your finding about the relationship between the two variables.

Task 4

1.  Construct a contingency table between the gender and the level of education. Using information in this table, can we say that male and female heads of the households differ in their higher level of qualification?
2.  What is the probability that the head of household is a female and her higher level of education is Intermediate?
3.  What is the probability that the head of household is a male and has the Bachelor degree?
4.  What is the proportion of having the Secondary as the highest degree from among females?
5.  Do you think that the events "gender of household head is male" and "having the Master Degree" are independent?

Answers:

Task 1

1. The given dataset contains information about 2000 households. A random sample of 200 households have been selected from the dataset using the technique of simple random sampling. Sampling has been done randomly as this is the most appropriate sampling technique that can be applied in this situation. There are certain advantages of this sampling technique for which this technique has been selected to be used for sampling. The advantages are given as follows:

• In this technique, the sample units are selected randomly.
• Each unit that is selected randomly for the purpose of the study has an equal probability of being selected in the sample.
• This type of sampling technique can be used with the availability of very little information about the population.
• This type of sampling technique is free from classification errors.
• The sample selected using this technique represents the whole population.
• The sample that has been selected is free from any kind of sampling biases.
• The analysis of data becomes much easier after sampling using this technique.
• The sampling technique can be used very easily as the technique is extremely simple.
• The errors in sampling becomes much more easy to assess by the use of this method.

1. Descriptive statistics for the annual expenditure of alcohol, meal, fuel and phone along with the comparison of these measures using a box and whisker plot are given in the following tables 1, 2, 3 and 4 and also in figure 1 respectively.

Figure 1: Box and whisker plot for annual family expenditure of Alcohol, meals, fuel and phone

Standard deviation is one of the best measures of variation. Standard deviation measures the deviation of the values of a variable from the mean of the variable. By comparing the standard deviations from tables 1, 2, 3 and 4, it can be seen that the highest standard deviation of the annual expenditure is on alcohol consumption. The standard deviation is found to be 2093.185. The standard deviation on the annual expenditure of meals is found to be 1577.824. The annual expenditure on fuels has a standard deviation of 1655.23 and the annual expenditure on phone is 1755.425. Thus, it can be said that expenditure on alcohol varies the most. Thus, the expenditure on alcohol by the families are not near the mean expenditure but are way from the mean expenditure. This can be less than or more than the mean value for the families. Among the expenses on these four necessities, meals show the least variability. Thus. The expenses are closer to the mean expense on meals.

From the box and whisker plot in figure 1, it can be seen that alcohol has the highest standard deviations. The least expense on alcohol by the families is zero. Thus, there are families who do not spend on alcohol. Again, there are values which are higher than the mean expense. Thus, there are families who spend quite high on alcohol. The expenses on meals are the least variable as this is a necessity in the life of people. Thus, it can be expected that higher income groups will spend more on meals and the lower income group will spend less but there is a minimum limit below which a person cannot spend on meals. Again, a person cannot have unlimited food. Thus, there is an upper bound too above which the expenses cannot rise. Thus, this explains the reason for minimum variability. The mean of all the expenses are higher than the median which is again higher than the modal expenditures. Thus, the expenses of most families are higher than the mean expenses.

Task 2

1. The annual expenditure on utilities has been summarized using the following frequency table:

Table 5: Frequency Table for Annual expenditure on Utilities

1. The following percentages have from the frequency table given in table 5:

• The percentage of households spending a maximum of $1200 per annum is 73.50%
• The percentage of households having expenditure on utilities between $1200 and $2400 per annum is (97 – 73.50) % = 23.50%.
• The percentage of households spending more than $2400 per annum is (100 – 97) % = 3%.

2. The distribution of the expenses of household on utilities is shown with the help of a histogram in figure 2.

Figure 2: Annual expenditure on Utilities by the households

1. From the histogram, it can be said very clearly that the expenses on utilities are not normally distributed. The expenses are skewed to the left. This indicates that most people have an annual expenditure on utilities which are less than the mean expenditure on utilities.

Task 3

Figure 3: Scatterplot of Annual after tax income and total expenditure

Table 6: Correlation Table

From figure 3 it is very clear that the natural log of the annual income and total expenditure depends on each other. The increase or decrease in total income indicates the rise or fall in the total expenditure respectively. The correlation between the two variables is 0.69 which indicates that there is a moderate positive relationship between the variables.

Task 4

Table 7: Frequency table of Gender and Highest degree of Education

Table 7 shows that the number of males undergoing higher level of education is 110 and the number of women undergoing higher level of education is 90. These two variables have very close values. Thus, it can be said that there is not much significant difference in the level of qualification in males and females.

1. Table 8 shows the percentages of gender and Education level

Table 8: Percentages of Gender and Education level

The probability that the head of the household is a female and her higher level of education is intermediate is (26/200) = 0.13.

1. The probability that the head of the family is a male and has bachelor’s degree is (21/200) = 0.105.
2. The proportion of having secondary as the highest degree from among females is ((23+23)/200) = 0.23.
3. Two events are said to be independent if the product of the probabilities of the two events separately are equal to the probability of the occurrence of the two events together. Thus,

The probability of gender of a household head being male is (90/200) = 0.45.

The probability of having a master’s degree is (41/200) = 0.205.

The probability of a household head being male and having a master’s degree is (15/200) = 0.075.

Now, (0.45*0.205) = 0.092 which is not equal to 0.075.

Thus, the two events are dependent.