Assignment 2 Inferential Statistic PDF

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ECON1193 – BUSINESS STATISTICAssignment 2 – Individual Case Study – Inferential StatisticTopic 2: Age dependency ratio, old (%working-age population)Student’s name Mai Minh NgocStudent ID SLecturer: Pham Thi Minh ThuyCampus: RMIT Hanoi Summarize descriptive statistic. In the previous assignment, I u...

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ECON1193 – BUSINESS STATISTIC Assignment 2 – Individual Case Study – Inferential Statistic Topic 2: Age dependency ratio, old (%working-age population)

Student’s name Student ID Lecturer: Campus:

Mai Minh Ngoc S3730089 Pham Thi Minh Thuy RMIT Hanoi

1. Summarize descriptive statistic. In the previous assignment, I used Descriptive Statistic to conclude about the ratio of old-age dependency (% working-age population) of 2 nations that Brazil and Kenya. This shows an evidence in the tables: Central Tendency

Mean

Median

Mode

Brazil

8.103964641

7.552690108

#N/A

Kenya

5.275461043

5.251903511

#N/A

Table 1: Central Tendency of Brazil and Kenya Coefficient of Country

Range

Interquartile Range

Variance

Standard Deviation

Variation

Brazil

5.376663755

2.605952993

2.650233071

1.627953645

20.088%

Kenya

1.92380989

0.97702102

0.35234497

0.593586531

11.25%

Table 2: The measures of Variation of Brazil and Kenya From two tables above about the age dependency ratio, old in Brazil and Kenya from 1980 to 2016 in Assignment 1. This assignment answered the topic question is “Is the age dependency ratio and income are related?”, which the answer is Yes. We know that, Brazil have higher income than Kenya, and furthermore, all the number of Brazil much hinger than Kenya about the Mean, Median, Variance, Standard Deviation and CV. With the table 3, all the percentage of age dependency and income always depend to each other. The higher income, the higher ratio of old age dependency and vice versa.

P(HI)

0.4

P(HI|LO)

0

P(HI|HO)

1

P(MI)

0.457142857

P(MI|LO)

0.761904762

P(MI|HO)

0

P(LI)

0.142857143

P(LI|LO)

0.238095238

P(LI|HO)

0

Table 3. The probability of the percentage age dependency and GNI. Word count: 134 2. Confidence intervals. a. Calculate confidence intervals. In this section, I’m going to calculate about Confidence Interval of age dependency ratio, old (% working-age population), gross national income (GNI) per capita (current US$) and life expectancy at birth, total (years) of 35 countries. The Sample size of Dataset 2 given is 35 countries, n = 35. In this part 2a, all calculation will choose Significant level at 5%, � = 0.05 → confidence level is 95% (1 – � = 0.95). In this part, we know about the information that: -

n = 35

-

population standard deviation � is unknown →

So, I’m going to use t-table. o Substitute the sample standard deviation, S o n = 35, degree of freedom, d.F = n - 1 = 35 - 1 = 34. o Lower, Upper tail =

→

α 2

=

0.05 2

= 0.025

tn-1, (α)⁄2 = 2.0322

Because of σ is unknown, so confidence interval estimated by: �=

´X ± tn-1

S √n

t is the critical value of the t distribution, o d.F = n-1

o Each tail = 

α 2

Age dependency ratio, old (% working-age population) o n = 35 o d.F = n – 1 = 35 – 1 = 34 o tn-1, (α)⁄2 = 2.0322 →

�=

´X + tn-1

�=

´X - tn-1

S √n

= 16.121 + (2.0322).

10.280 √35

=

19.652 S √n

= 16.121 - (2.0322).

10.280 √35

= 12.590

12.590 ≤ � ≤ 19.6522 As a result, we are 95% confident that the true mean age dependency ratio, old is between 12.590 and 19.652. This suggest that we are 95% sure that the ratio of elder (% working-age population) in the world in 2015 is between 12.590% and 19.652%. 

Gross National Income (GNI) per capita (Current US$) o n = 35 o d.F = n – 1 = 35 – 1 = 34 o tn-1, (α)⁄2 = 2.0322 →

�=

´X + tn-1

�=

´X - tn-1

S √n

= 19594 + (2.0322).

23429.862 √ 35

=

27642.262 S √n

= 19594 - (2.0322).

11545.737 ≤ � ≤ 27642.262

23429.862 √ 35

= 11545.737

We are 95% confident that the true mean of GNI per capita (current US$) is between 11545.737 and 27642.262. This is suggesting that in 2015, we are sure that 95% average person in the world earn US$ 11545.737 and US$ 27642.262. 

Life expectancy at birth, total (years) o n = 35 o d.F = n – 1 = 35 – 1 = 34 o tn-1, (α)⁄2 = 2.0322 →

�=

´X + tn-1

�=

´X - tn-1

S √n S √n

= 73.328 + (2.0322).

= 73.328 - (2.0322).

9.098 √ 35

9.098 √ 35

= 76.453

= 70.202

70.202 ≤ � ≤ 76.453 As a result, we are 95% confident that the mean of life expectancy at birth of the world in 2015 is between 70.202 and 76.453. This suggest we are 95% sure that an average newborn expectancy in the world in 2015 is between 70.202 and 76.453 years.

b. Discuss, what assumption are required to calculates these confidence intervals. The sample size of dataset 2 given is 35, n = 35 is large enough (n ≥ 30), we can apply Central Limit Theorem and sampling distribution for the mean can be approximated by normal distribution. Hence, we can use the normal distribution and the t-table without any assumptions of the population.

c. Discuss the possible impact on the confidence intervals, if the sample size increase double According to the formula of confidence interval is �1 =

´X ± tn-1

S . (with n1 = n) √n

With confidence interval �1 =

´X

The new confidence interval is �2 = Width confidence interval �2 =

´X

- tn-1

S √n

´X ± tn-1 + tn-1

´X + tn-1

-

S √2 n

S √n

= 2.(tn-1

S ) √n

S . (with n2 = 2n) √2 n -

´X + tn-1

S √2 n

= 2.(tn-1

S √2 n

) If the sample size (n increases, the standard error

S √n

will decrease because (n) is

a denominator. The data is more concentrated around the mean. The width confidence interval (± tn-1

S ) will decrease √n

√ 2 times. Therefore, the confidence

intervals will be more accurate. Interpretation: Increasing the sample size mean we are studying 70 nations instead of 35. Hence, the result of the world ratio of old-age dependency is represent better of all 195 countries in the world. A 95% CI, n=70 will be narrower than a 95%, n=35. Therefore, we can easily predict more accurately about the percentage retired people (% working-age population) in the world. Word count: 174.

3. Hypothesis testing. a) According to the report from United National the world average old age dependency ratio is 12.9% in 2013. The data from the report of United National shows that the average percentage of elder dependency in 2013 is 12.9% (%working-age population). However, the calculation in part 2 shows that the average of age dependency ratio, old is about 16.121% (%working-age population). It’s much higher than the percentage in 2013, (16.121% ˃ 12.9%). Therefore, the ratio of the world’s elder dependence expected in the future will increase. Age dependency ratio, old (%working-age population)

Mean

16.12118821

Table 4: The mean of age dependency ratio, old (%working-age population). Word count: 67. b) Hypothesis testing. 

Choosing Confidence level is 95% → level of significant (�) = 0.05.



No assumption required because sample size given is 35, n = 35 is large enough (n ≥ 30), so we can apply Central Limit Theorem. We can be using Ztable or t-table.

-

We called the Null Hypothesis is H0, and the Alternative Hypothesis is H1. The age dependency ratio, old of the world in 2013 is 12.9%. 

H0: ≤ 12.9 (old age dependency ratio is less than or equal to 12.9%).



H1: ˃ 12.9 (old age dependency ratio is more than 12.9%). → This is one tailed test – upper tail.



Since the population standard deviation, � is unknown → we are using t-table: The test statistic we use is: t =

o t= 

´ X−μ S √n

=

16.121 −12.9 10.280 √ 35

´ X−μ S √n = 1.853

(1)

� = 0.05 because we are using one tailed test, so the � will present for the upper rejection region. o d.F = n – 1 = 35 -1 = 34. o � = 0.05. → t n-1,

α

= 1.6909 (the critical value of t is 1.6909). (2)

(1) > (2), as the test statistic falls into in the rejection region. (t = 1.853 > 1.6909) → Hence the Null Hypothesis (H0) is rejected.

Interpretation: As reject H0, hence with 95% level of confidence it can be concluded that the age dependency ratio of old people of the world will not decrease in the future. 

As H0 is rejected, hence type I error might have been committed. →

� = P(Type I) = 5%

o Despite the test result, there are still have 5% of chance in the future, the ratio of elder in the world might decrease more than 12.9%. 

Minimizing type I error, we need to increase the confidence level = (1 – �) to decrease level of significant (�). Critical value will be more extreme and lager than the test statistic (CV > 1.853), hence we do not reject H 0.

4. Regression analysis. a) Determine the dependence variable (DV) and the independent variables (IV) of the dataset 2. 

Dependence variable (DV) o Age dependency ratio, old (%working-age population).



Independent variable (IV) o Gross Nation Income (GNI) per capita (current US$). o Domestic general government health expenditure per capita, PPP (current international $). o Life expectancy at birth, total (years). o Smoking prevalence, total (age 15+).

b) For each independent variables (IV), the relationship between DV and IV. b.1) Age dependency ratio, old (%working-age population) & Gross National Income, per capita (current US$). o Expected relationship: I expected to see a positive linear between age dependency ratio, old (DV) with the GNI per capita (IV).

Age dependency ratio, old (%working-age population)

Age dependency ratio, old & GNI per capita 40 35

f(x) = 0 x + 9.62 R² = 0.57

30 25 20 15 10 5 0

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

GNI per capita (US$)

Figure […]. The scatter plot of age dependency ratio, old & GNI per capita (US$). o Comment: According to the scatter plot, there is a little week linear relationship between the ratio of elder dependency and GNI per person; (0...