THE COVER PAGE BUSINESS STATISTICS individual case study sai gon south campus

Probability: Population Division n.d concluded that adolescent fertility rate is the annual number of births to per 1000 women who have ages from 15 to 19.. High Adolescent fertility rat

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BUSINESS STATISTICS

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I Introduction:

The adolescent birth rate is the measurement of the number of births per years from

1000 women who aged 15-19 years (WHO 2010)

Teenagers are more likely to face the danger during child-bearing period than mature women or even at risk of death and dis Besides, the newborns of teenage mothers are

at higher risk of low born weight and mortality (Measure Evaluation) In addition, the adolescent fertility also leads to the poor socioeconomic outcomes, due to the school

dropout, low productivity and poverty (Mcqueston, K & Silverman, R & Glassman, A 2012) Due to these reasons, raising the awareness as well as education in sex through sexual and reproductive medical care services is extremely essential (The United Nations)

With the high GNI countries and the developing countries such as United States, China and Europe would possess a lower ratio of teenage pregnancy countries

because they have taken advantaged from demographic dividend, so the reproduction have significantly decrease in the past few years In contrast, countries with lower GNI would possess a higher rate in reproduction, particularly Asia and Africa Based

on these evidences, we can see that in terms of poor countries, young mothers usually receive low income from unstable works or even have not enough qualifications as well as experience to raise their children (Guillaume 2016) Meanwhile, the reduction

in ratio of adolescent fertility in developed countries, which also illustrates that the environmental and well-being society are much better for children

II Descriptive Statistics and Probability:

1 Probability:

Population Division (n.d) concluded that adolescent fertility rate is the annual number of births to per 1000 women who have ages from 15 to 19 Then, the measure which is more than 30 births per 1,000 women ages 15-19 is considered high adolescent fertility rate On the other hand, different GNI level will classify different income level of that country:

Low-Income countries (LI): countries with a GNI less than $1,000 per capita

▪

Middle- Income countries (MI): countries with a GNI between $1,000 and $12,500 per

▪

capita

High-Income countries (HI): countries with a GNI greater than $12,500 per

▪

The studied countries in the provided data set would be divided into three categories based

on their GNI condition and the level of adolescent fertility

High Adolescent fertility rate (H)

Low Adolescent fertility rate (L) Total Low-income

Middle-income

High-income

Table 1.Contingency table for country categories in terms of income and the rate of adolescent fertility.

To determine if income level and mean annual adolescent fertility rate are statistically independent or not, conditional probability of related variables from two categories must be considered

P(H) = 20

35 = 4

7

P(H\LI) = P ( H∧LI )

P (LI ) =

8 35 8 35

= 1 ≠ P (H)

P(H\MI) = P ( H ∧MI )

P (MI ) =

12 35 16 35

= 3

4 ≠ P (H)

P(H\HI) = P ( H ∧HI )

P (HI ) =

0 35 11 35

= 0 ≠ P (H)

⇒ The categories countries above which are low, middle, and high income and high adolescent fertility rate are not statistically independent events

P(L) =

15

35 =

3 7

P(L\LI) = P ( L∧LI )

P (LI ) =

0 35 8 35

= 0 ≠ P (L)

P(L\MI) = P ( L∧MI)

P (MI ) =

4 35 16 35

= 1

4 ≠ P (L)

P(L\HI) = P ( L∧HI )

P (HI ) =

11 35 11 35

= 1 ≠ P (L)

⇒ The categories countries above which are low, middle, and high income and low adolescent fertility rate are also not statistically independent events

a The probability of country categories are more likely to have high teenager fertility rate:

P(H\LI) = P ( H ∧LI )

P (LI ) =

8 35 8 35

= 1 = 100%

P(H\MI) = P ( H∧MI )

P (MI ) =

12 35 16 35

= 3

4 = 75% P(H\LI) > P(H\MI) >

P(H\HI)

P(H\HI) = P ( H ∧HI )

P (HI ) =

0 35 11 35

= 0 = 0%

⇒ Countries with low income possess the highest in teenage birth rate

 Interpretation: As the independence between the country categories and low or high

adolescent fertility rate, it shows that the countries with high income have the lower adolescent fertility rate whereas the low and middle income countries may face the risk of the higher level in teenager pregnancy

2 Descriptive statistics:

a Measures of Central Tendency:

Low-income countries (births)

Middle-income countries (births)

High-income countries (births)

Table2.Measures of Central Tendency for the adolescent fertility rate

 Key findings:

- The average teenager pregnancy rate of low-income countries is the highest compared with that of middle-income countries and high-income countries (105.43 > 47.19 > 10.21)

- Similarly, the median of middle-income countries and high-income countries is also lower than that os low-income countries ( 8.76 < 47.34 < 98.88)

 Analysis:

In term of the Central Tendency of this case, the median is the most suitable compared with others The first reason is the distance between minimum values and maximum values is too far, which means the values of dataset have significant variation The major evident supporting the median measure is that all country categories have one outlier According to Jim, F (2019), mean is drastically impacted by outliers,

specifically, outliers would pull mean away from the center towards the longer tail As

a result, median is the better measurement to represent the central tendency for the distribution

b Measures of Variation:

Low-income countries (births)

Middle-income countries (births)

High-income countries (births)

Table3.Measures of Variation for the adolescent fertility rate

 Key findings:

- All the measures of low-income countries is higher than these of middle-income countries and high-income countries

 Analysis:

- In general, the standard deviation is considered as representative of the measurement

of variation Standard deviation does not include the middle values, implying that it not takes into account the distribution of data Range measure is also affected by outliers With IQR, it does not observe all the values in the data set, it just simply calculates the distance between Q1 and Q3 Variance is a squared unit, then, it does not provide the clear interpretation The final measure is CV, CV is also regarded as a good measure, however, it is better when the difference distribution of values is drastically big In short, the best measure in this case is Standard deviation

c Box and Whisker plot:

Low-income countries (births)

Middle-income countries (births)

High-income countries (births)

Whisker 17.89 < 54.86 17.07 < 53.26 1.59 < 10.32

Median 29.89 < 74.16 34.81 < 63.88 3.45 < 13.68

Table4.Box and Whisker plot

III Confidence Intervals:

1 Calculation:

In this case, the confidence level would be randomly chosen at 95% in order to estimate the confidence interval

In term of the sample size is 35 which is higher than 30, associating with the applying Central Limit Theorem (CLT) and the sampling distribution becomes normally distributed Because the population standard deviation ( σ¿ is unknown, the sample standard deviation (S) is substituted, and the t-table would be used

Population standard of deviation ( σ¿ Unknown births

Sample standard of deviation (S) 41.91 births

Table5.Statistics Summary for the adolescent fertility rate

 Calculate confidence interval:

α2 = 0,025 ⇒ t = ±2.0322

d.f = n-1 = 35-1 = 34

μ = X +t( S

√n)

⇒ μ =48.88 ± 2.0322(41.91

√35 )=48.88 ±14.4

⇒34.48 ≤ μ ≤ 63.28

 Interpretation: Regarding with 95% of confidence, we can claim that the world

average rate of adolescent fertility is between 34.48 births to 63.28 births per 1000 women

2 Assumption: Regardless of the lack of the population standard deviation, there is no

requirement for assumption As the sample size is 35 which is higher than 30, Central Limit Theorem (CLT) can be applied and the distribution would be normally

distributed

3 Supposing that the population standard deviation of adolescent fertility is known,

indicating that all the values of the population is collected and population mean would

be easier to calculate

From the equations to calculate the interval confidence ( μ= X +Z ( σ

√n) and

μ = X +t( S

√n) ), the main difference of the population standard deviation existence

is the application of the z-value instead of t-value Looking at the figure 1, we can see that the t-value is slightly bigger than z-value when the sample size is small

Therefore, if the critical value is larger, confidence interval would become wider In other words, if we use population standard deviation to calculate, the confidence interval would decrease, however, the accuracy would be boosted

Figure 1 Adapted from A Guide to Business Statistics (McEvoy 2018).

IV Hypothesis testing

1 According to The World Bank (2020), during the 18-year period, the mean of

adolescent fertility rate saw a rapid decrease from 61.335 births per 1000 women

(1996) to 48.36 births per 1000 women (2008), and the figure gradually go down to

46 births per 1000 women in 2014.In addition, based on the calculated confidence interval above (34.48 ≤μ ≤ 63.28) , assuming that the average rate of adolescent

pregnancy phenomenon would have tendency to decline in the future Therefore, Hypothesis Testing would be used to check whether this figure decrease or not

Figure1.Measures of Variation for the adolescent fertility rate

Hypothesis Testing (Critical Value approach)

Step 1: Check for Central Limit Theorem (CLT).

Since the sample size (n=35) is larger than 30, it is applicable to use CLT and the sampling distribution is normally distributed

Step 2: Determine hypothesis { The null hypothesis H0; μ ≥ 46

The alternativehypothesis H1; μ <46(claim )

Step 3: Based on the alternative hypothesis above, H1 contains ‘<’, thus lower-tailed test would be used

Step 4: Because there is no population standard deviation ( σ¿ provided and the sampling distribution is normally distributed, we use the T-table

Step 5: Determine critical value

Level of significance α=0.05

Degrees of freedom d f=34

Lower −tailed test }⇒ t=−1.69

Step 6: Calculate test statistic

t '=X −μ

S

√n

=48.88−46

41.91

√35

=0.41

Step 7: Make statistical decision.

Based on the graph above, the test statistic falls in Non-Rejection Region, as the

result, we do not reject H0

Step 8: Explanation

Regarding with 95% level of confidence, we can claim that the global average rate of adolescent fertility would significantly decrease

Step 9:

However, Type II error, P(Type II) = α = 0.05 = 5% might have happened when

we failed to reject a false null hypothesis (Trinh, T N 2020) Which means that most countries tend to decrease adolescent fertility ratio, whereas, there is still 5% of countries have growth in this phenomenon

2 Supposing the number of countries in dataset decrease by half, which leads to:

 The sample size is smaller (n<30) and the degrees of freedom (d.f) would go down

As the result, the t distribution would be at farer position compared with the position

of standard normal distribution That means, the smaller the sample size is, the less identical these distribution shapes

Figure2.Comparing t and Z distribution (Adapted from JMP)

 According to Chris Deziel (2018), the large sample size would increase the precision

of the testing In addition, the Type II error would be more likely to be committed

when the sample size is too small Then, the smaller sample size would have opposite effect compared with the larger one , namely, it would decrease the power of the study

⇒ Consequently, reducing the sample size reduces the confidence level of the study, which is related to the Z distribution The decline in the sample size also boosts the probability of error commission

V Overall conclusion:

After executing the research and calculating all the inferential statistics, we can conclude 3 core elements about the linking of countries income level and the adolescent fertility rate

Firstly, the income level is strongly associated with the adolescent fertility rate That means the countries with GNI less than 1000 per capita will face the higher risk of births, for example, the average births rate of low-income nation is 105.43 births In contrast, the birth rate of middle-income is approximately equal a half of that of low-middle-income (47.19 births) Especially, regarding to the high GNI nations would have extremely low adolescent fertility average rate, only 10.2 births Secondly, there is significant reduction in the teenager pregnancy ratio during the period from

1996 to 2014 To be more specific, there is 95% level of confidence that the rate would decrease significantly from 62.38 births to 34.48 births In addition, stating that 99% adolescent fertility rate would decrease at 44 births in 2014 and would keep on reducing in the coming years Lastly, looking at contingency table in part 1, the adolescent fertility rate totally is affected by the GNI 100% low-income countries have high ratio of teenager pregnancy phenomenon, whereas, the middle-income and high-income countries take the lower ratio, at 85% and 0% respectively

In short, the lower income level the nations are, the higher rate in adolescent fertility that they would get

To make the predictions become reality, we should improve sexual education for teenager Government should extensively propagate the disadvantages of pregnancy in the early age, such

as reducing life expectancy, having difficulty in raising children People should take this problem into consideration and together improve it to make our future more sustainable

VI References:

McEvoy, DM 2018, A guide to business statistics, John Wiley & Sons, Inc., Hoboken, New Jersey

Mcqueston, K & Silverman, R & Glassman, A 2012, ‘Adolescent Fertility in Low and

Middle-Income Countries: Effects and Solutions’, SSRN Electronic Journal.

Measure Evaluation 2017, Adolescent fertility rate, Measure Evaluation, viewed 10 December 2020

Population Division n.d, Adolescent Birth Rate, Population Division, viewed 8

December 2020,

<https://www.un.org/en/development/desa/population/publications/dataset/fertility/ad olescent-rate.asp#:~:text=Definition,for%20women%20aged%2015%2D19.>

The World Bank 2020, 'Adolescent fertility rate (birth per 1000 women ages

15-19), The World Bank, viewed 11 Decembe2020, <

https://data.worldbank.org/indicator/SP.ADO.TFRT>

Trinh, T N 2020, ‘Inferential Statistics – Hypothesis Testing’, PowerPoint slides, ICON1193B, RMIT university, Vietnam

United Nations 2014, Goal 3: Ensure healthy lives and promote well-being for all at all ages, United Nations, viewed 10 December 2020

World Health Organization 2018, Adolescent pregnancy, World Health Organization, viewed 9 December 2020