Part II descriptive statistics and probability

WORDs COUNT: 2,469 Part I: INTRODUCTION Sustainable development has become a major topic around the world, with the primary goal of improve well-being by better society, economic, and environment, aiming to develop lives for current and next generation (Dalby & Horton & Mahon 2019) Among 17 sustainable goals, in the 2015 framework for sustainable development goals, new-born survival and well-being is the priority to achieve the SDG 3, with social and health experts adapting a variety of statistical tools to assess the effectiveness of the healthcare system and the global health status in order to make improvements (World Health Organization 2020) Neonatal mortality rate (per 1,000 live births) is the ratio of deaths during the first 28 days of life, which is the feeblest period of a child’s time, in a specific time In the first month of life, children face the highest risk of dying (UNICEF 2020) Global newborn mortality has decreased from 36 fatalities per 1000 live births in 1990, according to UNICEF In 2019, the birth rate dropped to 17 per 1000 people Global newborn mortality has decreased from 5.0 million in 1990 to 2.4 million in 2019 The reduction in neonatal mortality is significant as part of the United Nations' Sustainable Development Goals since health is a fundamental goal of the SDGs because it ensures healthy lives and well-being for people of all ages In order to develop a strong workforce and meet the SDGs' goals The goal is to get neonatal mortality down to 12 deaths per 1000 live births Targeting the neonatal mortality ratio is critical, but precise measurement of the ratio remains a huge issue for a variety of reasons, including the fact that many deaths go unrecognized and uncounted due to the lack of good healthcare systems in low- and even in middle-income nations As a result, significant progress has been made in the decrease of neonatal mortality, there are still ways to improve and problems to overcome in the global community For many years, the relationship between national income and overall newborn mortality has been known A significant negative relationship between national income level and child mortality has been widely documented in a number of researches that analyze the relationship between change over time using both cross-sectional and trend data It was discovered that a 10% increase in GDP was connected with a 4% reduction in child mortality (Neal & Falkingham 2014) Part II: Descriptive Statistics and Probability: A: Probability: There are 34 countries in the data set 3, which are separated into types of income based on their Gross National Income (GNI) from World Bank in 2017 Low-income (LI) Middle-income (MI) High-income (HI) GNI less than $1,000 per capita GNI between $1,000 and $12,500 per capita GNI more than $12,500 per capita In addition, they are divided into two categories based on their neonatal mortality rates If there are more than 15 deaths per 1,000 live births, the nation is classified as having a high neonatal mortality ratio (H) Nations with a low neonatal mortality ratio if the countries have less than 15 fatalities per 1,000 live births (L) The following is a contingency table that categorizes countries into the following groups: Low-income countries (LI) Middle-income countries (MI) High-income countries (HI) Total High neonatal Low neonatal Total mortality rate (H) 11 15 mortality rate (L) 10 19 20 10 34 Table 1: Table Contingency of Neonatal Mortality Rate in three types of countries (unit: per 1,000 live births) 1: Statistical Dependence: The purpose of the contingency table is to determine whether the rate of income and neonatal mortality ratio are statistically dependent or independent, I will use a mathematical tool that is to compare between probability of all countries and high neonatal mortality ratio (H), as P (H), and a conditional probability of middle-income countries (MI) that have high neonatal mortality rate (H), P (H | MI), there is survey: P (H | MI) = P( H ∩ MI ) P(MI ) = 11 20 ≈ 0.55 => P (H | MI) P (H) = H Total countries = 15 34 ≈ ≠ P (H) 0.44 From the result before, a probability of countries having high neonatal mortality ratio is different from probability of high-income countries that have high neonatal mortality rate, which lead to that Gross National Income and Neonatal mortality rate are statistically dependent events To conclude from calculation of probability, the rate of income in each country can directly affect to the newborn mortality rate B: Descriptive Statistics 1- Outliers: Outlier Low-income (LI) Middle-income (MI) High-income (HI) Observation Value < Q1-1.5*IQR Or Observation value > Q3+1.5*IQR No Available Observation Value < Q1-1.5*IQR Or Observation value > Q3+1.5*IQR No Available Observation Value < Q1-1.5*IQR Or Observation value > Q3+1.5*IQR No Available s Result Table 2: Test of outliers in three types of income countries (units: deaths per 1,000 live births) Aiming to ensure that the analysis of descriptive statistics is accurate, the outliers should be found so the test table is conducted It is obviously seen that, there is no outliers available in trio of three categories of income countries It will be guaranteed that the outliers cannot be distorted the measurements of descriptive statistics 2- Central Tendency: Mean Median Mode Low-income (LI) 33.55 31.85 No Available Middle-income (MI) 18.30 16.30 No Available High-income (HI) 2.50 2.65 No Available Table 3: Central Tendency in three types of countries on Neonatal Mortality Rate (2017) (unit: deaths per 1,000 live births) Since the research in outliers shows that there is no extreme value in the trio of three categories countries, in this case, the most suitable measure of central tendency is Mean The reason for this statement is that Mean is an average of all values in data sets which is easily affected by the outliers When extreme values are unavailable, the Mean is the most reliable value to use As can be seen from table 2, the mean of low-income countries is the highest number with 33.55 deaths per 1,000 live births, which means that number is the largest number of all measures in the table The following is the mean of middle-income countries which is accounted to 18.30 deaths per 1,000 live births and high-income countries with the lowest mean with 2.50 deaths per 1,000 births Therefore, when compare between the rate of income and between the neonatal mortality rate, the lower income is, the higher neonatal mortality rate is It means that the high-income coutries saw a lower rate of newborn death 3- Variation Range Interquartile Standard Deviation Variance Coefficient of Variation Low-income (LI) 13.10 7.7 6.19 38.26 18.44% Middle-income (MI) High-income (HI) 29.70 4.00 18.33 1.55 10.27 1.21 105.40 1.46 56.10% 48.34% Table 3: Measure of Variation in three types of countries in Neonatal Mortality rate (2017) (unit: deaths per 1,000 live births) There is no extreme value available exist in all countries, it will be good to use all the values in a data set The Standard Deviation (SD) is considered as the best measure of Variation The mean of the distribution is used as a reference point for standard deviation The standard deviation shows the dispersion of values around the middle number From table 3, high-income countries have a lowest SD with 1.21 deaths per 1,000 live births, it shows that the data in this data set disperse around the average number Both low-income and middle-income countries have higher SD, which accounted for 6.19 and 10.27 deaths per 1,000 live births, respectively These figures indicate that the newborn mortality rates in two country categories differs significantly from the mean than the high-income countries Part III: Confidence Intervals 1- Calculation: This calculation conducted in order to show the confidence intervals of the average of the neonatal mortality rate (per 1,000 live births) It is assumed that the level of significance (α) is 0.05 Therefore, its confidence level is calculated: 1- 0.05 = 0.95 A table of data for confidence intervals is appeared below: Significance level Confidence level Population standard deviation Sample standard deviation Sample mean Sample size α (1 - )*100%  S ´ X n 0.05 95% unknown 12.65 15.45 34 As population standard deviation () is not available, I will use the sample standard deviation, hence, the students’ t-table will be used instead of z-table: Degree of freedom: d.f = n-1 = 33  t =  1.69236 Significant level: α = 0.05 Confidence intervals: μ = X´ ±t n−1 S = 15.45 ± 1.69236* √n 12.65 √ 34 11.78 ≤ μ ≤ 19.12 With 95% of confidence, the true mean neonatal mortality rate in 2017 ranges between 11.78 and 19.12 deaths per 1,000 live births 2- Assumption of Confidence Intervals: In the calculation of confidence intervals, sample size of data set is 34 that is larger than 30, a regulation of Central Limit Theorem (CLT) Hence the CLT is usable and the simple size is large enough to have a normal distribution Therefore, there is no assumption needed to calculate these confidence intervals 3- Confidence Intervals result when population standard deviation known: When the population standard deviation  is known, the z-table will be utilized because there are population standard deviation and sample size larger than 30 according to Central Limit Theorem In a student-t distribution, the mean and sample standard deviation are likely to change substantially from one sample to the next, resulting in a large degree of uncertainty in statistical work (Rumsey n.d) Because a confidence interval is a means to indicate what the uncertainty is with a certain statistic, it gets narrower as the degree of uncertainty decreases At the same time, when the sample size is small, critical z-values are lower than critical t-values for any degree of confidence When critical values are less, the width of confidence intervals will be narrow, confirming our hypothesis that using a population standard deviation causes shorter confidence intervals With the reduction in the width of confidence intervals, a precise result from the computation will be assured (Seb n,d) To conclusion, in the case of population standard deviation is available and is used instead of sample standard deviation, the standard deviation values will be reduced as a result of this The width of confidence intervals will narrow as the standard deviation decreases The confidence intervals will also be more accurate if the population numbers are used Part IV: Hypothesis Testing 1- Prediction of World Neonatal Mortality Rate: a- Comparing with the confidence intervals: According to World Health Organization (2016), the world average neonatal mortality rate is 18.6 deaths per 1,000 live births From the result of part 3.1, we are 95% confident that the world average newborn mortality ratio range from 11.78 to 19.12 deaths per 1,000 live births It is obviously seen that the population mean in the year 2016, 18.6 lies between 11.78 and 19.12 deaths, it is difficult to predict if the newborn mortality ratio will change or stay the same in the future Comparing with the sample mean, a point estimate of the confidence intervals is 15.45 deaths in 2017, which is lower than 18.6 deaths in 2016 As a result, I believe the world newborn death rate will fall in the future b- Hypothesis Testing: Significance level Confidence level Population standard deviation Sample standard deviation Sample mean Population mean Sample size α (1 - )*100%  S ´ X  n 5% 95% unknown 12.65 15.45 18.6 34 Step 1: Check for Central Limit Theorem: The size of data set is 34 which is higher than 30, the CLT is applied Step 2: Null and alternative hypothesis: Null hypothesis: H :  ¿ 18.6 Alternative hypothesis: H1 :  ≥ 18.6 Step 3: Determining the tailed: As the sign of H is “ ≥ , we would use the one tail- upper tailed test Step 4: Determining table: As the population standard deviation is unavailable and the CLT is applicable, we should use the t-table Step 5: Determining Critical Value (CV): Degree of Freedom: d.f = n-1=34-1=33  Critical value: t = +1.69236 (Upper tailed test) Significance level: α = 0.05 Step 6: Calculating test statistics: X´ − ¿ s t’= √n ¿ = 15.45 −18.6 12.65 √ 34 ≈ -1.45198 Step 7: Statistics Decisions: Figure 1: t-distribution graph Critical value lies on the non-rejection region as -1.45198 (t’) < 1.69236 (t) Therefore, we not reject H0 Step 8: Managerial conclusion in the context of Neonatal Mortality rate: As we not reject H0, therefore with 95% level of confidence we can state that the world average neonatal mortality rate will go down in the future Step 9: Determining the type of error: When we not reject H0, it will be possible that we might have committed Type II error (β) We can conclude that the newborn mortality ratio could not increase in the future, but in some chances, the rate of neonatal mortality will climb up afterwards The probability of making a type II error is equal to one minus the test's power The test's power can be raised by increasing the sample size, and vice versa, lowering the chance of making a type II error (Hayes 2021) 2- Possible impact on the hypothesis testing results when the data set decrease: In the case that the number of countries in the data set become half, if we want to research that is in the hypothesis situation, we need a certain sample size which helps us easily define whether it is in Type I or Type II error In which we reduce the sample size then it might have impacts on the hypothesis testing The tscore and sample size have a positively correlated relationship, when reducing in sample size, t-score also decreases while the size decline, it leads to the margin of error increases, confidence intervals will increase (Rumsey n.d) On behalf of the value of t-scores, we can state that the statistical decisions might change when the sample size decrease In our case, when the critical value t decrease, the test statistics t’ also decreases, leading the point t’ cannot reach the Reject Region as the Figure illustrated Hence, in the data set 3, I would say that although the sample size becomes half, the statistical decision will stay unchanged The case of sample size declines, the accuracy of the result will be lower With this increase in error, the statistic result becomes less accurate as of the bigger the marginal error gets (Rumsey n.d) Also increasing the sample size can increase the Power of the Test, with a inverse case, smaller sample size means less information we have, and the error raise, thus boost a 5% chance of committing Type II error (Rousseau 2003) Part V: Conclusion: In conclusion, this report on the data set of 34 countries related to the neonatal mortality ratio has stated that newborn mortality rate and Gross National Income (GNI) has a negative correlation Moreover, the neonatal mortality ratio is speculated to witness a descending in the future I will point out some factors to describe about the relationship between the Neonatal Mortality rate (NMR) and Gross National Income (GNI) First of all, in part II.A, NMR and GNI are dependent factors, meaning that one country has high or low newborn mortality ratio based on the income per capita of that nation In particular, countries with a high level of income are less likely to have a high neonatal mortality ratio, supported by the probability of countries having a high newborn mortality rate given that they are low-income being the highest NMR and GNI have a significantly negative correlation is proved in part II.B, the average number of deaths is falling significantly from 33.55 deaths to only nearly 2.50 deaths per 1,000 live births when the GNI rises, state that the higher national income is, the lower neonatal mortality ratio a country has It concludes that high-income nations have the lowest NMR, owing to a comparably small number of neonatal fatalities, as mentioned in Part II, whereas middle and low-income countries have significant higher NMR, bolstering the assertion of a negative relationship between NMR and GNI From part III, the result is that the average world neonatal mortality rate is range from 11.78 to 19.12 deaths per 1,000 live births in 2017, with the report of World Health Organisation, the world average newborn mortality is lies between the range before accounting to 18.6 deaths per lives From part IV, we 95% confidence that the neonatal mortality rate will decrease in the future, but with 5% that the newborn mortality ratio will increase In light of the foregoing, these findings may prove useful in strategic planning for achieving global SDG target 3: once the relationship between NMR and GNI is fully understood, experts now recognize that, in addition to lowering mortality rates, we must improve living standards and healthcare systems They should focus their efforts on the low and some middle-income nations and regional issues such as health care, social, and education, while also monitoring sophisticated healthcare functions to the neonatal survival and also healthcare for mother Following that, suitable steps to assist low-income nations in preventing newborn mortality would be implemented Reference: Dalby, S, Horton, S & Mahon, R 2019, Achieving the Sustainable Development Goals : Global Governance Challenges, Taylor & Francis Group, Ebook Cetral Hayes, A 2021, Type II Error Definition, Investopedia, viewed 19 August 2021, Neal, S & Falkingham, J 2014, 'Neonatal Death and National Income in Developing Countries: Will Economic Growth Reduce Deaths in the First Month of Life?', viewed 18 August 2021 Rousseau, B 2003, Power of a Test , Encyclopedia of Food Sciences and Nutrition (Second Edition), , ScienceDirect, viewed 18 August 2021, Rumsey, D n.d, How Sample Size Affects the Margin of Error, Dummies, viewed 19 August 2021, Rumsey, D n.d, How to Calculate a Confidence Interval for a Population Mean When You Know Its Standard Deviation - dummies, dummies, viewed 18 August 2021, Seb n.d, Confidence Intervals and Z Score, PROGRAMMATHICALLY, viewed 19 August 2021, UNICEF 2020, Neonatal mortality, UNICEF, viewed 18 August 2021, World Health Organization 2020, Newborns: improving survival and well-being, viewed 20 August 2021,  ... connected with a 4% reduction in child mortality (Neal & Falkingham 2014) Part II: Descriptive Statistics and Probability: A: Probability: There are 34 countries in the data set 3, which are separated... as mentioned in Part II, whereas middle and low-income countries have significant higher NMR, bolstering the assertion of a negative relationship between NMR and GNI From part III, the result... (Seb n,d) To conclusion, in the case of population standard deviation is available and is used instead of sample standard deviation, the standard deviation values will be reduced as a result of