Business_Maths_&_Stats

BMS11: Business Maths & Statistics a) The statistical summary table from Part A returned the following results from the salary observations in the Environmental Policy survey: Salary Summary |   |   |   | Male | Female | Mean | $ 53,695 | $ 58,643 | Median | $ 50,050 | $ 58,100 | Stdeviation | $ 10,201 | $ 13,731 | Minimum | $ 37,700 | $ 31,000 | Maximum | $ 78,000 | $ 81,400 | Range | $ 40,300 | $ 50,400 | 1st Quartile | $ 45,400 | $ 49,600 | 3rd Quartile | $ 62,100 | $ 62,000 | Inter-quartile range | $ 16,700 | $ 12,400 | 80th Percentile | $ 62,760 | $ 69,760 | Coefficient of variation | 19.00% | 23.41% | | | | Average salary is higher for female respondents based on the mean of $58,643 against $53,695 for males. As this data is ordered/ranked XXX a more appropriate measurement of the centre of the distribution is obtained from the median. Again, this is higher for females, but of more interest is the relationship between mean and median. The female observations have very similar measures of central location, indicating that the distribution is symmetric. If this lack of parity in the male data were greater it would indicate that there are either extreme observations in the data set or there is a substantial skew in the data. In addition to identifying the central location, we need to consider how typical that average value is of the set of measurements, and consider the variability, or spread of observations around the average. In these two sets, the range of the female data is greater as demonstrated by both the Range calculation (determined by deducting the lowest observation (MIN) from the largest observation (MAX) and the variance (average of squared deviations from mean to measure dispersion). The distribution of data sets display bell shaped properties so we can apply the empirical rule (that most (95%) of the sample observations will fall within 2 times the standard deviation from the mean) to check that the calculated values are reasonable, and confirm that for both sets of data the range/4 is approximately equal to the calculated standard deviation: Male Range/4 = $10,075 Std Dev = $10,201 Female: Range/4 = $12,600 Std Dev= $13,731 Extrapolating the empirical rule, we can observe that the male salary measurements are more clustered around the mean - 68% of male observations lie between $43,494-$63,896, against $44,912-$72,374 for females. The observations for 95% and 99.7% reflect the observed spread of measurements for both sets. This clustering in the male observations reinforces the observed distribution showing a clear modal class. Further to the consideration of variability, the coefficient of variation is calculated to identify whether the standard deviation value is large or small – simply returning a number value does not give an indication against the rest of the sample. The coefficient of variation is the standard deviation as a percentage of the mean, and for both data sets it is approximately 20%, although higher for the female data set. The interquartile range(IQR) of the data sets present an interesting observation. The IQR measures the spread of the middle 50% of observations, and is particularly useful for ranked data such as these salary observations and this measure is not sensitive to extreme or outlier values. The male data set returns a larger IQR measurement than the female set ($16,700 against $12,400), meaning that the first and third quartiles are further apart, and indicates a greater degree of variability. Preparing a histogram of the two data sets showing relative frequency distribution across equal class widths reveals the following information: (Red line is approximation of Median, Green line is approximation of Mean) * The centre of the histogram for female salaries is higher than that for males – as evidenced by higher values for both median and mean measures. * The spread of salaries for females is greater than males, as evidenced by the range of the female observations having both a lower minimum value and higher maximum value. * The female salary histogram appears generally symmetric with a bell shaped distribution, however the lack of observations in the $75,000 class would not produce a frequency polygon with symmetric or bell shaped characteristics. In fact the chart is somewhat negatively skewed with the larger proportion of the observation occur in the higher end of the salary scale. In contrast, the male histogram is also unimodal and shows a more dominant modal class (returning a relative frequency of 45. The ‘shape’ of the histogram is positively skewed with the number of observations trailing in the higher classes. b) The median salary of females is $58,100. This is the value that falls in the middle of the ordered set of measurements. In this sample, the number of responses is an even number, therefore the median is the average of the two middle observations: Median = 58100+58100/2 = $58,100. As the salary measurements are ordered (ranked), the median should be the preferred measure of central location. In the female sample there are no extreme observations to distort or skew the mean, so these two measures of central location are quite closely aligned at median : $58,100 and mean: $58,640. This is compared to the parity between the median and mean values of male salaries in question d) below. c) The sample data presented does not support the statement that on average, males are older than females in this survey. Although the sample size of female respondents is smaller (Females: n=44, Males : n=58), the mean value for age is equal for both the male and female respondents at 42 years of age. At two decimal places, the mean value for the female respondents is slightly higher at 42.89 years (against 42.74 years for males). d) On the basis of the mean and median values for salaries, what can be said about the shape of the salary distributions of male and female respondents' As noted in question b) above, the measures of central location for salary from the female respondents are closely aligned at median : $58,100 and mean: $58,640. This should indicate a symmetric distribution, with half the observations falling above and half below the mean. In comparison, the median value of the male salary data is $50,050 against a mean of $53,695. This is not a significant variation on its own given the range of values but is notable in comparison to the parity of the mean and median values of the female respondents. This suggests that the male values may contain an outlier or extreme observation in the higher end that is distorting the mean. This distribution is unlikely to be symmetric due to the lack of correlation between the mean and median values and the lack of a consistent measure of central location. e) In the data set provided, the variable for age is quantitative (numerical) while the variable for opinion is categorical (nominal). Quantitative (or numerical) values are, as the same suggests quantitative measures or values that must be real numbers and not representative of categories - therefore the data for age is quantitative. Categorical or nominal observations are arbitrary values assigned to represent the categories of responses. In the assignment example, the environmental policy survey respondents gave an opinion in response to Likert scale groupings of: strongly disagree/disagree/neutral/agree/strongly agree and a numerical value of 1 -5 was assigned to each category for the purposes of undertaking analysis of the frequency of responses across the sample (as frequency calculations are the only valid statistical interrogations o nominal data).