Statistical Notes for Clinical Researchers

Question: Discuss about the Statistical Notes for Clinical Researchers. Answer: Introduction: The annual rainfall and climate of the country Singapore has been analyzed in this report. The country Singapore is located in tropical rainforest region. The country is surrounded by sea on all its sides. The region experiences maritime climate all over the year. There are no distinct seasons. The country also experiences moderate temperature and huge amount of rainfall all over the year. The average temperature of the country is about 88 degree Farenheight. The country is located to the south of the Equator. December and January are the coldest month while the April and May are the hottest month (Xavier et al. 2014). In this report, the climate of the country Singapore is being studied. The temperature, humidity, rainfall of the country is being analyzed using various statistical techniques. The report will help to get an idea about the overall climate of the country. The calculations, charts and analysis for this report has been done using excel software. The main research question considered for this report is: Is there any relationship between the temperature and humidity? Is there any relationship between the sulphur di oxide and nitrogen di oxide content and air temperature? Is there any relationship between the rainfall and bright sunshine? Data Analysis, results and discussion: The descriptive statistics measures helps to get an idea about the dataset. The summary statistics measures include the measures of central tendency, dispersion, histogram and other graphs and charts. The summary statistics measures have been calculated for the variables amount of rainfall, sunshine, average temperature and others. Temperature(Max) Sunshine Relative humidity Total rainfall Sulpher di oxide content Nitrogen di oxide content Mean 31.42857 Mean 5.614286 Mean 82.42857 Mean 2184.571 Mean 11.42857 Mean 23.71429 Standard Error 0.114879 Standard Error 0.105624 Standard Error 0.739921 Standard Error 150.606 Standard Error 0.649437 Standard Error 0.521641 Median 31.3 Median 5.6 Median 82.9 Median 2159.9 Median 11 Median 24 Mode 31.2 Mode #N/A Mode #N/A Mode #N/A Mode 11 Mode 25 Standard Deviation 0.303942 Standard Deviation 0.279455 Standard Deviation 1.957647 Standard Deviation 398.4661 Standard Deviation 1.718249 Standard Deviation 1.380131 Sample Variance 0.092381 Sample Variance 0.078095 Sample Variance 3.832381 Sample Variance 158775.2 Sample Variance 2.952381 Sample Variance 1.904762 Kurtosis -1.40432 Kurtosis -0.76392 Kurtosis 2.95556 Kurtosis 0.043713 Kurtosis -0.63809 Kurtosis -2.0895 Skewness 0.567796 Skewness -0.01178 Skewness -1.5093 Skewness -0.25177 Skewness 0.168964 Skewness -0.35866 Range 0.8 Range 0.8 Range 6.1 Range 1210 Range 5 Range 3 Minimum 31.1 Minimum 5.2 Minimum 78.5 Minimum 1538.4 Minimum 9 Minimum 22 Maximum 31.9 Maximum 6 Maximum 84.6 Maximum 2748.4 Maximum 14 Maximum 25 Sum 220 Sum 39.3 Sum 577 Sum 15292 Sum 80 Sum 166 Count 7 Count 7 Count 7 Count 7 Count 7 Count 7 Table 1: Descriptive statistics measures (Source: Created by author) The mean value of the average daily maximum temperature of Singapore is found to be 31.42857. The mean temperature has a deviation of 0.303942. The median value of the temperature is 31.3 degree Celsius and the modal value is 31.2 degree Celsius. The skewness value is 0.567796 and the range of temperature has a negative kurtosis measure. The range of maximum temperature very the years is however very small only 0.8. The summary statistics measures states that the daily Sunshine has a mean value of 5.614826. The mean value has a deviation of 0.279455. The median value of Sunshine is 5.6. The Sunshine value is different for different years from 2008 to 2014. Therefore, there is no modal value of Sunshine. The range of Sunshine over the years is 0.8. The Sunshine variable has almost symmetric distribution the value of skewness being only -0.01178. The country experiences a maritime climate and therefore is expected to have a high value of relative humidity. The average value of relative humidity all over the years is 82.42857 and the median is 82.9. The average value has a deviation of 1.957647 from the mean value. The distribution of relative humidity is negatively skewed indicating that some of the years have a low value of relative humidity. The entire range of relative humidity is 6.1. The mean value of total rainfall of the country over the year is 2184.571 which have a deviation of 398.4661 from the mean value. The skewness value of the distribution is -0.25177. The entire range of observation is 1210. The median value of total rainfall is 2159.9 (cm) which indicates that there is no such outlier present in the observation. The sulpher di oxide content has a mean value of 11.42857 which has a deviation of 1.718249. The entire range of observation is 5 and the distribution is positively skewed. The mean value of Nitrogen di oxide content is 23.71429 and the median value is 24. The modal value of nitrogen di oxide content is 25. The mean value has a deviation of 1.380131. The range of the observation of nitrogen di oxide content is 3. The distribution of sulpher di oxide content is negatively skewed. Discrete random variable: The number of days in te year having rainfall is a discrete random variable. The number of days in the year can take the values 1, 2, 3,........ Out of 365 days in a year, the number of days in the year that is expected to have rainfall is said to follow a binomial distribution. The average number of days in the year that is expected to have rainfall is 180.4286. The proportion of days when the rainfall occurs is 0.493151 or is approximately 0.5. Therefore, a binomial distribution has been fitted by using the parameters n = 365 and p = 0.5. The probability distribution function of the variable is given below: The above graph plots the values of number of rainy days along x axis and the CDF and the PDF values of along the y axis. The PDF values are shown with the help of the red line. The red line shows that the distribution of PDF is approximately symmetric. The CDF of the distribution gradually increases and reaches a maximum value at 1. Inferential statistics: The distribution is said to be normally distributed if the distribution of the values are symmetric. The mean, median and modal values of the normal distribution coincide. The value of skewness for the normal distribution is zero. The value of kurtosis measure for the normal distribution is also zero. The values of the normal distribution are expected to lie within 3 limit. About 99% of the values of normal distribution lie within the 3 limit. The variable average temperature is expected to follow a normal distribution (Kim 2013). The deviation from normality could be best measured with the help of Kurtosis value (Bates et al. 2014). The skewness value for the variable is 0.56776. Therefore, the value is much less and the variable can be assumed to be normally distributed. A test of hypothesis has been conducted for the variable temperature. The average daily maximum temperature of Singapore is 31 degree Celsius. A hypothesis test has been conducted to test this claim. The null hypothesis of the test is H0: = 31 and the alternative hypothesis is H1: 31. The test statistic is given by: t = (x-bar )/(/sqrt(n)). The test statistic is said to follow a normal distribution. As the number of observation is very small, less than 30, the population standard deviation has been approximated by the sample standard deviation. The value of the test statistic is 3.73062. The value of the tabulated z at 5% level of significance is 1.96 for both sided test. The null hypothesis of the test is rejected on the basis of the calculated value of the test. The average temperature cannot be assumed to be 31 degree Celsius. Correlation: The relative humidity and the temperature have some kind of relationship between themselves. As the value of temperature increases, the value of relative humidity decreases and vice versa. The relative humidity is actually a ratio between the pressures of water vapor at a given temperature to the pressure of water vapor at equilibrium. When the temperature is low, the relative humidity is high and when the temperature is high, the relative humidity is low. The Pearsonian correlation has been calculated for temperature and relative humidity. The value of the variable is -0.39935. Therefore, there is a negative correlation between the two variables. The correlation has also been calculated for the variables Sunshine and total rainfall. The rainfall and Sunshine are negatively related. The value of the correlation coefficient has been calculated to be -0.77086. This means as the value of Sunshine increases, the value of rainfall decreases. Regression: The air temperature has a relationship between the Sulpher di oxide content and the Nitrogen di oxide content. The relationship among the variables has been explored with the help of regression analysis. A linear regression line has been fitted for the variables. The regression line obtained from the equation is given below: y = 33.22166 -0.0235 * Sulpher di oxide content 0.06429* Nitrogen di oxide content. The regression equation states that there is a negative relationship between sulpher di oxide content and nitrogen di oxide content and the air temperature. The air temperature decreases as the nitrogen and sulpher di oxide content increases. The R-squared for the regression analysis is 0.151147. Therefore, the model is not a very good fitted model in reducing the error and predicting values from regression equation. The summary statistics measures like mean, standard deviation, variance, skewness and kurtosis measures are being calculated for each of the variables. A test has been conducted to know about the average temperature of the country. The average temperature has been found to be not equal to 31 degree Celsius. The relationship between the air temperature and relative humidity is being studied. It has been found that there is a negative relationship between the temperature and relative humidity. The Sunshine and rainfall is also said to follow a negative relationship. The air temperature is inversely related to the nitrogen di oxide and suplher di oxide concentration in the air. The increase in sulpher di oxide and nitrogen di oxide content decreases the air temperature as is indicated by the results of regression analysis. Conclusion: The report gives an idea about the overall climate of Singapore. Various climates related issues are being studied in this assignment. The country Singapore is located in South of the Equator in tropical region. The country is surrounded by sea on all its sides. The country experiences very high temperature and huge amount of rainfall all over the year. The relative humidity in the country is very high as well (average 82%). The Sulpher di oxide and Nitrogen di oxide content in the air has a negative relationship with air temperature. References: Bates, D., Maechler, M., Bolker, B. and Walker, S., 2014. lme4: Linear mixed-effects models using Eigen and S4.R package version,1(7). Blanca, M.J., Arnau, J., Lpez-Montiel, D., Bono, R. and Bendayan, R., 2013. Skewness and kurtosis in real data samples.Methodology. Draper, N.R. and Smith, H., 2014.Applied regression analysis. John Wiley Sons. Kim, H.Y., 2013. Statistical notes for clinical researchers: assessing normal distribution (2) using skewness and kurtosis.Restorative dentistry endodontics,38(1), pp.52-54. Kim, H.Y., 2013. Statistical notes for clinical researchers: assessing normal distribution (2) using skewness and kurtosis.Restorative dentistry endodontics,38(1), pp.52-54. Regezi, J.A., Sciubba, J.J. and Jordan, R.C., 2016.Oral pathology: clinical pathologic correlations. 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