(TIỂU LUẬN) the impact of the halved sample size on hypothesis testing results

Subject Code: ECON1193 Subject Name: Business Statistic Location & Campus: RMIT SGS Lecturer: Nguyen Thi Tuong Chau Title of Assignment: Individual case study on Inferential Statistics Dataset: Mortality rate – Dataset Student name: Nguyen Huu Nam Phuong Student ID: S3892189 Word count: 2666 Page: 10 Assignment due date: 22/8/2021 Date of Submission: 22/8/2021 Table of Contents I Introduction: II Descriptive Statistics and Probability: .3 A Probability: Test of independence Probability B III Descriptive statistics: Measurement of Central Tendency Measures of Variation Confidence Intervals: A Calculation: B Assumptions: C The impact of the World Standard Deviation on Confidence Interval IV Hypothesis Testing: A Critical Value Approach: .8 B The Impact of the Halved Sample Size on Hypothesis Testing Results: 10 V VI Conclusion: .10 Reference: 11 I Introduction:  Global Description: The death of a newborn within the first month is referred to as neonatal mortality The United Nations International Children's Emergency Fund (UNICEF) reported the average global neonatal mortality rate of 17 deaths per 1,000 live births in 2019, or roughly 2.4 million children worldwide perished in the first month (UNICEF 2018) A progress compared to the world’s neonatal deaths of million in 1990 Despite the significant decline in mortality rate, WHO reported huge disparities in the neonatal mortality rate between high-income and low-income countries In 2019, neonatal mortality remains highest in Sub-Saharan Africa 27 deaths per 1,000 births, followed by Central and Southern Asia (World Health Organization 2019) In other words, newborns in these low-income countries are 10 times more likely to die than those in Europe or North America (World Health Organization 2019)  The importance of reducing the neonatal mortality rate Reducing the neonatal mortality rate is seen as an important target in achieving the UN Sustainable Development Goal 3: Good Health and Wellbeing which “[ ] promotes healthy lifestyles, preventive measures and modern, efficient healthcare for everyone” (The Global Goals 2016) The UN has designated the neonatal mortality rate as a public health indicator measuring children and communities’ access to medical treatments and nutritious diet (Global SDG Indicator Platform 2021) According to WHO, the main causes for neonatal mortality are preterm birth, intrapartum-related complications, infections, and birth defects which can be prevented through antenatal care, eradication of communicable disease, and skilled medical personnel (World Health Organization 2019) Experts projected that if the world does not accelerate its pace in reducing the neonatal mortality rate, then 1.8 million newborns will die by 2030 (Hug et al 2019) However, if most of the world achieved the newborn mortality rate of 12 per 1,000 births in each nation by 2030 then the number of neonatal deaths will be decreased to 1.2 million (Hug et al 2019)  The correlation between Gross National Income (GNI) and Neonatal Mortality rate Multiple studies have analyzed the impact of economic conditions on the level of infant mortality According to Prichett & Summers (1996), “the wealthier nations are the healthier nations” with the conclusion that poor economic performance in the 1980s contributed to the death of 0.5 million infants in low-income nations in 1990 alone Preston (2017) argues that citizens in higher NGI per capita nations have higher living standards as they spend more money on health-related goods and services (World Health Organization n.d.) While, in low-income countries, poverty may be the cause of malnutrition, and lack of access to education, housing, and health care, among other things that are necessary to improve health and lower mortality rates (Biciunaite 2014) II Descriptive Statistics and Probability: According to the World Health Organization (WHO), the Neonatal Mortality rate is “the number of deaths during the first 28 completed days of life per 1000 live births in a given year or other period” In this case study, a country is considered to have a high newborn mortality rate if the death rate exceeds 15 deaths per 1,000 live birth On the other hand, those that have lesser than 15 deaths per 1,000 live births are categorized as low neonatal mortality rate countries Additionally, the observed 33 nations are also categorized into three levels of income based on Gross National Income (NGI) per capita Neonatal Mortality rate High-Income (HI) Middle-Income Level of Income (MI) Low-Income (LI) Total High Low Total 12 12 14 19 2 26 33 Figure 1: Contingency table of countries in terms of NGI and neonatal mortality rate A Probability: Test of independence Two events are independent if their joint probabilities are equal to their individual probabilities (Brereton 2016) To evaluate the statistical independence of the country’s income level and newborn mortality rate, the conditional probability of the two variables is compared to the probability of a country with high mortality rate The probability of a country with high neonatal mortality rate, P (high neonatal mortality rate), is compared to the probability of a country with high mortality rates given that they are highincome countries, P (high neonatal mortality rate | HI) P ( High neonatal mortality rate|HI ) = P( Highneonatal mortality rate∩ HI ) =0 P(HI ) P(high neonatal mortality rate) = 0.21 Because P ( High neonatal mortality rate|HI ) ≠ P(highneonatal mortality rate) , High-income and high neonatal mortality rate are two statistically dependent events We can infer that there a country’s GNI is related to its newborn mortality rate Correspondingly, comparisons are also made with the probability of a country with high mortality rates given that they are Middle-or Low-countries, P (high neonatal mortality rate | MI or LI) � ( High neonatal mortality rate|MI ) = P ( High neonatal mortality rate| LI ) = P( Highneonatal mortality rate∩ MI ) P(HI ) = 0.26 P( Highneonatal mortality rate∩ LI ) =1 P ( HI ) P(high neonatal mortality rate) = 0.21 Because ( High neonatal mortality rate|MI ∨ LI ) ≠ P ( high neonatal mortality rate) , we can conclude that a country’s level of income and neonatal mortality rate are statistically dependent events Probability � (ℎ�� ℎ neonatal mortality rate ���� │� �) = or 0% � (ℎ��ℎ neonatal mortality rate ����│� �) = 0.26 or 26% � (ℎ��ℎ neonatal mortality rate ����│��) = or 100% In addition, the calculated probabilities above show that all low-income nations in this sample have high neonatal mortality rates Whereas there is a 0% chance a high-income country observed in the sample has high newborn mortality rate Middle-income nations rank second with 26.3% having high mortality rate In conclusion, wealthier countries are less likely to have high neonatal mortality rate as the newborn mortality rate is dependent on a country GNI B Descriptive statistics: To select the best descriptive statistics' measures, the data set categorized on nations’ level of income are examined for outliers High-income Middleincome Low-income Min 1.1 > Lower Bound -1.29 Max 8.9 > Upper Bound 6.81 Result outlier 3.3 > -6.73 43 > 32.68 1outlier > 8.2 33.7 < Figure 2: Table for identifying outliers 42.2 No outlier 16.7 Measurement of Central Tendency High-income Middle-income Low-income Mean 3.48 14.09 25.2 Median 2.8 11 25.2 Mode 2.8 N/A N/A Figure 3: Measurements of Central Tendency Table (deaths per 1,000 live births) As shown in figure 2, the data set has one outlier in each of the middle-and high-income groups The mean is the arithmetic average of all the values therefore sensitive to outliers (Frost 2019) The existence of outliers in the dataset would cause the mean to move closer to the outliers affecting the result of the analysis On the other hand, the median does not consider all the values but focuses on the middle values of the data set instead (Australian Bureau of Statistics 2009) Hence, the median is the most suitable measurement to examine the death rate of three income categories Moreover, the mode for high-and middle-income nations is undetected According to figure 3, low-income countries have the highest median of 25.2 deaths per 1,000 live births, twice the median value of the middle-income nations (11 deaths per 1,000 live births), and significantly higher than high-income countries (2.8 deaths per 1,000 live births) This further supports the argument that the probability of a nation having a high mortality rate depends on the level of income proven in the independence test Because the median of the newborn mortality rate in low-income nations exceeds 15 deaths per 1,000 births, we can infer that countries with low national wealth are most likely to have high and most extreme neonatal death rates On the other hand, the median of middle-income countries is 11 We can infer that half of the nations have higher death rates than 11 deaths per 1,000 births Moreover, the probability of those countries having high mortality rates is 26% earning the second spot, followed by high-income countries Measures of Variation Range IQR Standard Deviation Sample Variance Coefficient of Variation High-income 7.8 2.21 2.67 7.12 Middle-income 39.7 9.85 10.44 108.98 Low-income 17 8.5 12.02 144.5 76.58 74.09 47.7 (%) Figure 4: Measurement of Variation table (deaths per 1,000 live births) The measure of variation should not be susceptible to the dataset’s outliers Like the median, the Interquartile Range is not drastically affected by outliers (Frost 2019) Hence, the Interquartile Range is the most suitable measure of variability for this dataset giving information on the fluctuation of newborn mortality rate in different levels of national income As shown in figure 4, the Interquartile Range of middle-income countries is the highest (9.85) followed by low-income (8.5) and high-income (2.21) countries This indicates that the neonatal mortality rates in middle-income nations fluctuate greatly compared to those of the other two levels of income III Confidence Intervals: A Calculation: Significance Level Confidence Level Critical value Population Standard Deviation Sample Standard Deviation Sample Mean Symbol α (1- α) x 100% t n−1 σ S X´ Value 0.05 95% 2.04 N/A 10.38 10.91 Sample size n 33 Degree of freedom n-1 32 Figure 5: Summary table for the world average neonatal mortality rate 95% is the chosen confidence level to calculate the world average neonatal mortality rate Since the population standard deviation (σ) is unknown, its substitution is the sample standard deviation, S In addition, the t student distribution is utilized as the σ is unknown ´ ± t n−1 × μ= X S 10.38 =10.91 ±2.04 × √n √33 →7.22 ≤ μ ≤ 14.6 We are 95% confident that the true mean of the world average neonatal mortality rate will fall between 7.22 and 14.6 deaths per 1,000 live births B Assumptions: Despite the population standard deviation (σ) is unknown, no assumption is needed Since the Central Limit Theorem is applicable as the sample size is large enough (n>30), the sampling distribution is normally distributed C The impact of the World Standard Deviation on Confidence Interval According to Anderson (2013), the sample standard deviation is a statistic that varies according to the sample taken from the population Hence, the sample standard deviation has greater variability compared to the population standard deviation which is a fixed value parameter (Taylor 2019) The variability of the sample standard deviation creates a level of uncertainty when conducting statistical calculations (Anderson 2013) The result of statistical calculation would be more accurate if the standard deviation of the population is known In addition, the z-table would be used to find the critical value instead of the t-table In this case, the critical Z-value would be smaller than the t-value Since the tails of the t-distribution are shorter and fatter than the Z distribution, the t-standard deviation is larger than that of Z (Rumsey 2019, pp 106–108) A smaller Z-value would lead to a smaller critical value which in turn shorten the confidence interval and increase accuracy Thus, the use of the world standard deviation of neonatal mortality rate would reduce the confidence intervals In addition, this confidence interval of the global neonatal mortality rate is more accurate IV Hypothesis Testing: A Critical Value Approach: According to WHO, the world’s 2016 average neonatal mortality rate is 18.6 deaths per 1,000 live births In comparison with the 2017 calculated confidence interval ( 7.22≤ μ ≤14.6 ¿ , the world average mortality rate exceeds the range In addition, the point estimate of the confidence interval, the sample mean, (10.91 deaths per 1,000 births) in 2017 is lower compared to the 2016 mortality rate Therefore, the world newborn mortality rate is expected to fall in upcoming years Symbol Significance Level α Confidence Level (1- α) x 100% Population Mean µ Population Standard Deviation σ Sample Mean X´ Sample Standard Deviation S Sample Size n Figure 6: Summary table for the Hypothesis testing  Value 0.05 95% 18.6 N/A 10.91 10.38 33 Step 1: Central Limit Theorem The Central Limit Theorem applies as the sample size is large enough (n > 30) Hence, the sampling distribution is normally distributed  Step 2: Set up the hypotheses: The null hypothesis ( H ): The alternative hypothesis  μ ≥ 18.6 H (¿¿ 1) : � < 18.6 ¿ Step 3: Select the rejection region The chosen level of significance (α): α=0.05 Sample size (n): n=33 It is a lower-tailed test as the alternative hypothesis  H (¿¿ 1) contains “