ASIGNMENT 2 individual case study on inferential statistics in term of relationship between the age dependency ratio and GNI

In term of relationship between the age dependency ratio and GNI, United Nations 2015 measured that 22% of the population in high-income countries was older people while it occupies ab

ASIGNMENT 2: Individual Case Study on Inferential Statistics

Course Code: ECON1193B

Course Name: Business Statistic 1

Lecturer: Huy Doan Bao

Campus: Saigon South

Student Name: Le Cong Minh

Student ID: S3824313

World count: 2785

Numbers of pages: 8

1

Table of content

I Introduction 3

II Descriptive statistics and probability 4

2.1 Probability 4

2.2 Descriptive statistics 5

III Confidence intervals 6

a Calculation 6

b Assumption 7

IV Hypothesis Testing 7

V Conclusion 9

VI Reference 11

VII Appendices 12

I Introduction

In recent years, population aging has been recognized as a global phenomenon Beside the concern of population explosion, another inextricable issue is the imbalance in age

dependency, which is considered as the main reason why measures of age dependency ratio are extremely integral According World Bank (2020), age dependency ratio is defined as the ratio between dependents people (younger than 15 or older than 64) and the working age population (range from 15 to 64)

More tellingly, the meaning of dependency ratio is supporting economists by providing them clearly an overall change of population growth Additionally, it also shows the impacts of population ageing on society in several fields including productivity, disability and

dependence, social security sustainability and innovation due to the different characteristics

of old and young individuals (BMJ Global Health 2019) For a typical example, if the

dependency ratio is high, it means the number of retirement population will go up

substantially, which leads to the decrease of saving rates With the lower saving rates,

economic growth will be facing to the prevention ofinvestment rate since there will be less funding for investment projects Therefore, it will cause long-term economic changes

After conducting several academic researches, WHO (2011) stated that we will have more older people sooner or later and more people at extreme old age than ever before It means the fewer people of working age; the fewer people can support the society Last year, the total number of people over 65 was estimated around 700 million In addition, this global number

is predicted to reach over 1.5 billion people, which means will double the present number in over next 3 decades (Appendix 1) (United Nations 2019) It happens due to a remarkable improvement in life expectancy (WHO 2011) For instance, the rise of risk diseases such as: heart disease, cancer and diabetes lead to the changes in healthier lifestyle and diet as well as the rapid development of healthcare industry

The dominant mission of SDG3 is ensure healthy lives and promote well-being for all at all ages (United Nations 2015) so monitoring the ratio of age dependency certainly plays a crucial role For example, by observing the ratio, it is obvious that the number of older people keeps rising well while the proportion of children under 5 years old surviving decreases moderately each year since 1970s (Appendix 2) In order to reach one of SDG3 targets that ending all preventable deaths under 5 years of age, WHO and UNICEF has cooperated together worldwide to overcome this difficulty by providing nutrition, immunization, high-impact health, HIV and early child development interventions, as well as safe water and sanitation services in every region of the world, including in fragile and conflict settings (WHO 2020) They are taking attempt to ensure the chance of children survival, growth, and benefit from a safe and clean environment.

In term of relationship between the age dependency ratio and GNI, United Nations (2015)

measured that 22% of the population in high-income countries was older people while it

occupies about 13% for upper-middle-income countries and 5% for low-income countries

3

(Appendix 3 and 4) Due to the enormous growth of economy in developed countries, older

people have a higher level of consumption (Appendix 5) as well as better government

support and treatment such as transfer payments, saving rates, coverage pension, inactive

labour force than their peers in developing countries Consequently, it jumps to the

conclusion that high-income countries tend to be the most aged (United Nations 2015)

Overall, this paper purpose is illustrating the clear relation between the age dependency ratio

and GNI by applying descriptive statistics, inferential statistics, and probability to analyze the

data set of 35 countries

II Descriptive statistics and probability

2.1 Probability

United Nations (2020) concluded that the proportion of age dependency in a country, which is

greater than 20%, is regarded as the high level of age dependency ratio On the other hand, the

income levels of each country will be divided into 3 groups based totally on the GNI level.

Considering as a low-income country (LI), the GNI must be less than $1,000 per capita,

Considering as a middle-income country (MI), the GNI must be in the range between

$1,00 and $12,500 per capita

Considering as a high-income country (HI), the GNI must be greater than $12,500

per capita

According to the provided data (Appendix 6), the contingency table was drawn and characterized by

the combination between GNI condition and the level of age dependency ratio.

countries (LI)

countries (MI)

countries (HI)

Figure 1 Contingency table for country categories in terms of income

level and age dependency ratio

a To illustrate whether the income level and the age dependency ratio are statistically independent

or not, we must find out exactly the conditional probability of two relevant events from both two

categories by academic accounting In this case, it is obviously recognized that two integral events

will be high-income countries (HI) and high age dependency ratio (H).

P (H )=14

35

P (H|HI )= P(H∧HI )

= 35

=1

P(HI) 14

35

Due to the difference between P (H ) and P (H| HI ) ( 14 ≠1 ), high age dependency ratio

35 and high income is not considered as the independent events so one event will be surely affected by the probability of another one Thus, we can sum up that income and age

dependency ratio are statistically dependent.

b The ability of a country category does not have a high age dependency ratio can be

compared more clearly and informatively by accounting the probabilities of not high age dependency ratio with each income level

5

P (N ∨LI )=P (N∧LI)

= 35 =1=100 %

P(LI)5

35 16

P (N|MI)=P ( N ∧MI )

= 35 =1=100 %

35 0

P (N|HI )= P(N ∧HI )

= 35

=0=0 % P(HI) 14

35

P (N ∨LI )=P(N|MI)> P (N|HI )

Accordingly, the above calculations stated that all of countries with middle-income and low-income tend to have a low age dependency ratio with the same probabilities of 100%

whereas thanks to the improvement in life expectancy as well as modern healthcare system, country with high-income has become a safe home for elders and infants

2.2 Descriptive statistics

countries (LI) countries (MI) countries (HI)

Figure 2 Measurement of central tendency table of age dependency ratio of three country

categories (%)

Low outlier >,<,= Minimum values

5

High outlier >,<,= Maximum Figure 3 2 comparison tables between extreme

MI 17.77 > 15.75 As the results in Figure 3 show that outliers exist in

HI 37.8 > 34.99 this data set, Mean cannot be the most suitable

measure due to its sensitization to outliers

Furthermore, this data set contains only numerical data as well as Mode cannot be identified clearly so it will be rejected to be an ideal approach

at all In short, among all tendency methods, Median seems to be the most effective measure

in this case

The high-income category accounted for the highest median (28.35%) with all of units higher than

the standard level of high age dependency ratio (20%) Incidentally, this properly

14 appropriate to the probability calculation: P (H| HI )= 35

= 1 =100 % , which means 100% 14 35

of high-income countries will have the age dependency ratio more than 20% and is

possibly considered as a high ratio On the contrary, the median of both medium-income

and low-income countries’ age dependency ratio (9.1% and 5.05%) are the same lower

than the figure for high-income countries (28.35%) As this result, high age dependency

ratio is likely to occur in high-income countries rather than the rest

III Confidence intervals

a Calculation

In case of estimating the confidence intervals, the confidence level must be selected Due

to no requirement, I randomly choose 95% in this case

t-critical value (using online calculator) ± 2.03

Figure 4 Statistics summary table for the age dependency ratio

As the data set includes 35 countries, it means the sample size will be equal to 35

(n=35), which is greater than 30 Therefore, we will apply Central Limit Theorem

(CLT) for accounting as well as the sampling distribution is normally distributed

Beside that, by missing the population standard of deviation ( σ ), it results in the

substitution of sample standard of deviation (S) and the use of T-table.

μ=16.12 ± 2.03( 10

√ 35 .28 )=16.12 ±3.53

=> 12.59≤ μ≤19.65

Statement: we are 95% confident to jump to the conclusion that the world average of

age dependency ratio will fluctuate from 12.59% to 19.65%

Despite of the lack of population standard of deviation, any requirement for assumption

is not integral and become excessive as the sample size (n=35) is obviously greater

than 30 Hence, Central Limit Theorem (CLT) should be made use of accounting in

case of this normal sampling distribution

c. Because the world standard deviation of age dependency ratio is supposed to be known,

it means calculating the population mean as well as collecting all the values in the entire population can be conducted more easily Therefore, infernal statistics are not essential for the population mean estimation Additionally, there is a lack of major difference between sample to sample, which leads to no commitment for the standard error of the mean (Levine et al 2016)

According to the equations: μ=X+Z( σ ) and μ= X +t( S ) , it obviously shows

that instead of calculating t-value, z-value will be applied As the t-distribution shape

seems to be flatter than the z-distribution (Figure 5), the t-value is marginally greater

than the z-value in case of the small sample size (McEvoy 2013) The greater critical

value is, the larger confidence interval width will be so violating in the inverse

relationship: the narrower confidence interval width, the higher accuracy

Figure 5 Image captured from A Guide to Business Statistics (McEvoy 2018).

In short, if the world mean standard deviation of age dependency ratio is given, the

confidence interval will be decreased Otherwise, the result will become more exact.

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IV Hypothesis Testing

a. According to the World Bank (2020), during the time period time from 1990 to 2013, the world

average of age dependency ratio only continuously went up and reached the peak at 12.062% in 2013

(Appendix 7) Then, in the next year, WHO (2014) reported that this total number was up to 12.9% Based

on the above result of confidence interval (

12.59 ≤ μ ≤19.65 ), it is predicted that the global mean ratio of age dependency

will have an upward tendency in the future

Figure 6 Statistics summary table for world average of age dependency ratio

Hypothesis Testing Calculation (Critical Value approach)

Step 1: Verify the Central Limit Theorem (CLT)

As the sample size (n=35) is higher than 30, it leads to the normal sampling distribution

and application of CLT

{ The null hypothesis H0 :μ ≤ 12.9

Step 2: State hypothesis

The alternative hypothesis H1 : μ>12.9(claim)

Step 3: Based on the above alternative hypothesis ( H1 ), they contain ‘ ¿ ’ Therefore,

upper-tailed test would be applied in this case

Step 4: Due to missing the population of standard deviation as well as the conclusion

of a normal sampling distribution, T-table will be applicable for this calculation

Step 5: Define critical value

Level of significance α=0.05

}

Degrees of freedom d f =34 ⇒

t cv =1.69

Upper−tailed test

Step 6: Calculate test statistic

t stat

Step 7: Make statistical decision.

In case of the fall of test statistic into Rejection Region (t stat > t cv (1.85>1.69)) and the

calculated confidence interval in part III, H 0 will consequently be rejected while H 1 is acceptable at all

Step 8: Interpretation

Because of the result of confidence interval and support evidence from the rejection of H 0 by Hypothesis Testing, we could infer that the increase in mean of age dependency ratio will continuously boost in the future with 95% level of confidence

Step 9:

As H0 is rejected, it means we would commit Type I error to our analysis

With P (Type I) = α = 0.05 = 5%, which is regarded as the percentage of the chance that the mean of age dependency ratio could stop increasing in the future

In order to minimize the Type I error, decrease the significance level is the most common way Thus, in this case, the level of significance should be determined at 1% (0,01), which means only 1% probability of incorrectly rejecting the null hypothesis

b Double sample size

Even though the supposition of a double number of countries is carried out, there is still no any effects on the conclusion of the aforementioned hypothesis testing, apart from being more precise

Certainly, the rise in sample size (n) as well as degrees of freedom (df) are directly resulted from the double number of countries Hence, it leads to the change in t-distribution by moving gradually nearer

to standardize normal distribution Figure 7 shows that the more sample size

increase, the more homogeneous these distribution shapes will be. Then, if the sample size

9

seems to be large enough, t and Z distribution are likely to have no variance Consequently, the

accuracy of estimating the standard of deviation will be improved sharply (Berenson 2015).

Figure 7 Standardized normal distribution and t distribution for 5 degrees of freedom,

adopted from Basic Business Statistics eBook (Berenson 2015).

So as to gain a better evaluation, we should widen our knowledge about studied population dramatically by increasing the sample size based on the theory that the greater sample size is, the more margin of error in the confidence interval can be decreased (Bowerman, Froelich and Duckworth 2018) Moreover, the reduction of uncertainty and inaccuracy will be enhanced as the narrower confidence interval is, which caused by the increase in sample size, the more statistical power and greater precision can be achieved (Littler 2015)

After illustrating the relationship between GNI and the age dependency ratio by calculation and analysis on the summarized descriptive analysis, probability, confidence intervals and hypothesis testing, we gain some key findings from the above inferential statistics

First of all, by drawing the contingency table and calculating the probability, it shows that the level of income and the age dependency ratio are both dependent events, which leads to the GNI impact on the ratio of age dependency when some changes happen Due to the strong association between the income level and age dependency ratio, developed countries with high-income (GNI over $12500) level will trend in the growth rate of high age dependency steadily whereas in developing countries with low-income or medium-income level (GNI less

or equal to $12500), the welfare of older and infant population needs to be concerned in order

to lengthen the life expectancy as well as strengthen the survival rate

Next, it is reckoned that 100% of high-income countries will have the age dependency ratio higher than 20% On the other hand, the probabilities of medium-income and low-income countries having a high age dependency ratio are the same and equal to 0% Furthermore, the median of high-income category (28.35%) can be seen obviously the highest one from

descriptive analysis, meanwhile the lowest median (5.05%) belongs to the low-income

countries Beside that, the median of medium-income countries (9.1%) is extremely less than the critical value of age dependency level (20%) As this result, it is inferred that the higher income level, the higher age dependency will be

Last but not least, with the support evidence from confidence intervals as well as hypothesis testing, we are 95% confident to estimate that the world average age dependency ratio will continuously keep the upward trend in growing in the future Based on the above calculation, the mean age dependency ratio in 2015 (16.12%) is enormously greater than the 2014

(12.9%) Hence with 95% level of confidence, we can conclude that the world mean age dependency ratio will be volatile in the range from 12.59% to 19.65% Fortunately, the