Project II Data Analysis and Inference - Coursework Example

Running Head: Project II Data Analysis and Inference Project II Data Analysis and Inference Inserts His/her Introduction The given statistical report comprises the sample data selected from the survey conducted by U.S. Health Department. The sample data chosen for statistical examination includes the breadth of elbow in centimeters of male respondents. The sample size contains 40 individuals, particularly, male. The sampling design chosen for the given data is simple random sampling to avoid biasness in the results as it provides each unit equal opportunity to be chosen. The sample chosen by random sampling method is a representative of the population, so the sample results can be used to interpret the population results, such as on a level of significance it can be seen that the sample mean is an unbiased estimator of population mean. Moreover, the results drawn are free of any biasness. Different statistical measures including mean, median, mode, standard deviation, variance, range, quartiles etc., are calculated using the given sample data, and details are included in the given text. Median, Mode and Mean The central tendency measures help in selecting a single unit out of a sample as a representative of sample. The most commonly used measures include mean, median and mode. These measures calculated for the given data are as follows. Table 1: Mean, Mode and Median Measures of central tendency Mean 7.295 Median 7.3 Mode 7.5 These measures show that the middle value, calculated as median is 7.5cm, The most recurring value in the sample as shown by mode is 7.5cm and the average as calculated by mean is 7.295 cm i.e., around 7.3cm. Since the mean value and the median are approximately same, hence, both of them can be regarded as the best representative of the given sample. Dispersion Measures: The significance of the mean value to be the representative of the given sample can be assessed by using the dispersion measures. The range being the simplest measure of variation explains how distant the lowest value and the highest value in a data set are located. Variance and standard deviation, on the other hand, show how much each data point varies from the mean value calculated. For the given data set these are as follows. Table 2: Standard Deviation, Range and Variance Measures of variation Standard Deviation 0.451749 Variance 0.204077 Range 1.8 The standard deviation shows that each unit in the given sample varies by 0.45 from the mean value. The sample variance is 0.20 while the data range is 1.8 showing that the highest and the lowest value in the given sample differ by 1.8. However, the small value of standard deviation shows that the majority of the values lie near the mean value. The Box and Whisker Plot and The Five Number Summary The box and whisker plot is a tentative graphic is employed to explain the sample distribution at a glance. Before drawing a box plot, a five number summary that includes the following is prepared. Table 3: Five number summary Five statistical summary   First Quartile 7 Second Quartile 7.3 Third Quartile 7.5 Interquartile Range 0.5 Lower Limit 6.25 Upper Limit 8.25 BOX PLOT Q1 Q2 Q3 5 5.5 6 6.5 7 7.5 8 8.5 Q1 represents the first quartile, 7 in this case, showing that a quarter of the whole data set lies above this value. Q2 represents the median which divides the whole data set into two halves, so that half of the values lie above the median value and half lie below this value. Q3 represents the third quartile, 7.5 in this case, shows that 75% of the values lie above it while remaining 25% lie below the third quartile. The horizontal bars show the upper and lower limit, in this case, it is 8.25 and 6.25 respectively. Outliers are the values within the data set that do not lie in between the lower and the upper limits; in this case only one value 8.3 is the outlier, represented in the box plot as a circle. Moreover, the distribution of the given data set is left skewed as shown in the box plot and the middle values of the data set fall within the interquartile range (Triola, 2010). Histogram The following table was generated by MS Excel to prepare a histogram. Table 4: Frequency distribution and Histogram Bin Frequency 6.5 1 6.7-6.8 5 6.9-7.1 12 7.2-7.4 7 7.5-7.7 8 7.8-8.0 4 More 3 The histogram shows a double peaked distribution of the given data values. The Chebyshev’s theorem and The Empirical rule The Chebyshev’s theorem states that a fraction of data must lie within a particular number of mean’s standard deviation and is applicable to any data values regardless of its distribution shape. However, the empirical rule applies to a bell shaped, symmetric distribution (Peck & Devore, 2011). Table 5: Chebyshevs theorem and Empirical rule   Minus SD Plus SD Actual Count Percent Chebyshev’s Empirical Mean +/- one SD 6.8433 7.7467 28 70%   68% Mean +/- two SD 6.3915 8.1985 38 95% >=75% 95% Mean +/- three SD 5.9398 8.6502 40 100% >=85% 99.70% The given sample data satisfies the Chebyshev’s theorem as well as the empirical rule. According to Chebyshev’s theorem, more than 75% of the elbow breadth will be between 6.3915cm and 8.1985cm, while more than 85% of the elbow breadth will be between 5.9398cm to 8.6502cm. However, empirical rule shows that 68% of the breadth of elbow will be between 6.8433cm and 7.7467cm, 95% will be between 6.3915cm and 8.1985cm and 99.70% will be between 5.9398cm to 8.6502cm. Confidence Interval The rationale behind doing simple random sampling from a population and calculating sample statistics is to use that sample statistic for the prediction of population statistics. But the accuracy of such estimation is doubtful. However, constructing a confidence interval helps by providing a range of set values which are more likely to hold the parameter of population within consideration. These are constructed at a confidence level. For example, 95% confidence interval shows that if numerous samples are generated from the same population, then the sample statistics would be a approximately 95% a predictor of population parameter. The requirements for constructing confidence interval include selecting the confidence level and the sample statistic and calculating the margin of error. Table 6: Confidence Interval Confidence Level Alpha Margin of Error Lower Limit Upper Limit Elbow         90 0.1 0.1175 7.1775 7.4125 98 0.02 0.1662 7.1288 7.4612 In the given data, the sample mean was selected as an estimator of population mean. At 90% confidence level, the population mean is expected to lie in the 7.3±0.1175 interval. However, at 98% confidence level, the population mean is expected to lie in the 7.3±0.1662 interval. The confidence interval constructed at 98% level is larger due to the increased margin of error (Levine, 2008). Conclusion The statistical measures show varying results for the given data. The measures of central tendency showed median and mean of the sample to be approximately equal. The measures of variability showed that the each sample data unit varied from the mean by 0.45175. The Box and whisker plot showed that the data distribution was left skewed while the histogram predicted frequency distribution to be double peaked. The 98% confidence interval level showed that the population mean is expected to lie in the 7.3±0.1662 interval. References Levine, D. M. (2008). Business Statistics: A First Course. India: Pearson Education. Peck, R., & Devore, J. L. (2011). Statistics: The Exploration and Analysis of Data. (7th ed.). Cengage Learning. Triola, M. F. (2010). Elementary Statistics Using the TI-83/84 Plus Calculator. (3rd ed.). Pearson Education. Appendix A Table 7: Elbow breadth in cm, U.S. Department of Health and Human Services, Third National Health and Nutrition Examination Survey. Male Elbow 6.5 6.6 6.6 6.7 6.7 6.8 6.9 6.9 6.9 7.0 7.0 7.0 7.0 7.1 7.1 7.1 7.1 7.1 7.2 7.3 7.3 7.3 7.4 7.4 7.4 7.5 7.5 7.5 7.5 7.5 7.5 7.6 7.7 7.8 7.8 7.9 8.0 8.1 8.2 Appendix B The statistical analysis done to prepare this report helped me in grasping the basics of statistical measures including sampling methods, measures of central tendency, measures of variability, five number summary, box and whisker plot, etc., Moreover, it helped me in understanding the key concepts and increased my motivation to further endeavor in this subject. Read More