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Statistics And Research Methods, Holmes Institute, Australia

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Question 1: Australian exports (goods and services) along with its top 8 export markets in 2004-05 and 2014-15. Using this data, answer the questions below.

(a) Use an appropriate graphical technique to compare the value of Australian exports (in A$ bn) in 2004-05 and 2014-15, broken down by country of export destination.

Answer: 

Statistics And Research Methods 1.jpg

(b) Use an appropriate graphical technique to compare the percentage value of Australian exports (in %) in 2004-05 and 2014-15, broken down by country of export destination.

Answer:

Statistics And Research Methods 2.jpg

(c) Comment your observations in parts (a) and (b).

Answer: In part a), the highest export of goods and service in 2014-2015 was China the least in that season was United States recorded. In 2004-05, Japan recorded the highest export and the least was Singapore.

In part b) the highest in percentage value of exportswas China with 40.4% in 2014-15. United Kingdom recorded the lowest percentage value of exports in 2014-15. Japan recorded the highest percentage value of exports in 2004-05 and the lowest was Singapore.

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Question 2:

a. Construct a frequency distribution and a relative frequency distribution for the data.

b. Construct a cumulative frequency distribution and a cumulative relative frequency distribution for the data.

Answer:

Classes

frequency

Relative frequency

Cumulative frequency

Cumulative Relative frequency

30-40

2.00

0.05

2.00

0.05

40-50

4.00

0.10

6.00

0.15

50-60

8.00

0.20

14.00

0.35

60-70

11.00

0.28

25.00

0.63

70-80

8.00

0.20

33.00

0.83

80-90

5.00

0.13

38.00

0.95

90-100

2.00

0.05

40.00

1.00

 

40.00

     

c. Plot a relative frequency histogram for the data.

Answer: Histogram

Statistics And Research Methods 3.jpg

d. Construct an ogive for the data.

Answer:

Statistics And Research Methods 4.jpg

e. What proportion of the grades is less than 60?

Answer: 14/40 = 0.35

f. What proportion of the grades is more than 70?

Answer: 15/40 =0.375

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Question 3: a. Using an appropriate graphical descriptive measure (relevant for time series data) describe the two variables.

Answer:

Statistics And Research Methods 5.jpg

Statistics And Research Methods 6.jpg

The first graph of retail turn over per capita seems to be rising from 1983-2014. The second graph of final consumption expenditure is also in an uptrend since 1983-2014.

b. Use an appropriate plot to investigate the relationship between FINAL CONSUMPTION EXPENDITURE and RETAIL TURNOVER PER CAPITA. Briefly explain the selection of each variable on the X and Y axes and why?

Answer:

Statistics And Research Methods 7.jpg

c. Prepare a numerical summary report about the data on the two variables by including the summary measures, mean, median, range, variance, standard deviation, coefficient of variation, smallest and largest values, and the three quartiles, for each variable.

Answer: x- variable is retail turnover per capita and y-axis is Final expenditure.

d. Calculate the coefficient of correlation (r) between FINAL CONSUMPTION EXPENDITURE and RETAIL TURNOVER PER CAPITA. Then, interpret it.

Answer: Descriptive Statistics Return per capita

Mean

2205.762

Median

2180.2

Range

1558.7

Variance

295059.6

Standard deviation

543.1939

Co-efficient of Variation

24.63

Minimum

1455.9

Maximum

3014.6

1st Quartile

1641.8

2nd Quartile

2180.2

3rd Quartile

2799.7

Descriptive Statistics Final expenditure

Mean

146019.9

Median

139137

Range

151259

Variance

2.200

Standard deviation

46904

Co-efficient of Variation

32.12

Minimum

81889

Maximum

233148

1st Quartile

103169

2nd Quartile

139137

3rd Quartile

192828

Coefficient of Correlation

Coefficient of correlation indicate the strength of relationship between two variables.

Correlation coeff, r: 0.9876971

P-value (two-tailed): 0.000

The coefficient of correlation is 0.988 which means that there is a strong positive correlation between Return per capita and final expenditure.

e. Estimate a simple linear regression model and present the estimated linear equation. Then, interpret the coefficient estimates of the linear model.

Answer: Regression analysis show the relationship between variables(Patten & Newhart, 2017).

Linear equation is given by Y= B0 +BX1

Regression Results:

Y= b0 + b1x:

Y Intercept, b0: -42102.53

Slope, b1: 85.28681

The linear equation is Y= -42102.53 +85.29X1

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f. Determine the coefficient of determination R2 and interpret it.

Answer: The coefficient of determination R2

Coeff of Det, R^2: 0.9755456

Coefficient of Det R^2 is 0.976 meaning that 97.6% of the variation of the dependent variable (Final expenditure) can be explained by the independent variable (return in capital).

g. Test whether FINAL CONSUMPTION EXPENDITURE positively and significantly increases with RETAIL TURNOVER PER CAPITA at the 5% significance level.

Answer: The p-value 0.000< 0.05 appears to be significant.

h. What is the value of the standard error of the estimate (se). Then, comment on the fitness of the linear regression model?

Answer: Standard error is 7363.225.

The linear regression is a strong fit for the prediction.

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