Comprehensive Statistics - Assignment Example

Comprehensive statistics: Question There are four measurement level and they include: a. Nominal - in this level of measurement data is only givena label or name to represent categories, for example when players in a football game are assigned numbers, the numbers only and no rank is implied. Statistical calculations such as mean, median and standard deviations are meaningless. b. Ordinal - in ordinal level of measurement data are assigned names and can be ranked, for example data that represent workers job satisfaction will have information such as disagree, agree and strongly agree. Another example is the level of education attained where data may contain number 1, 2, 3, 4 and 5 where 5 represents high education attainment. Therefore nominal data where attributes can be ranked but the distance between ranks has no meaning. c. interval data- in this level of measurement the distance between ranks has a meaning, data can be ranked and at the same time the distance has meaning, for example temperature data is interval data where temperature difference has meaning. The average and median calculated has a meaning but ratio calculations have no meaning where one cannot state that 100 degrees Celsius is twice 50 degrees Celsius. d. ratio- in ratio level of measurement the absolute zero value has a meaning, all statistical calculation of central tendencies and dispersion are meaningful, for example data containing number of customers that visit a retail shop, the value zero has meaning and the ratio calculations also has meaning where it can be stated that 20 customers is twice 10 customers. Question 2 The following are the advantages and disadvantages of the bar chart, pie and Pareto diagram: Bar chart: Example: Advantages: The bar chart is appealing and easy to understand It is easy to compare categories Numeric values are provided for each category Disadvantages: Can only be used for discrete data type The graph can be reordered in order to show certain characteristics of categories. Pie chart: Example: Advantages: The pie chart is appealing and easy to understand The chart gives the percentages of each category Disadvantages: The pie chart does not provide numeric values for each category The pie chart cannot accommodate many categories Can only be used for discrete data type Pareto diagram: Advantages: It is easy to understand and helps in emphasizing important categories Helps identify the main causes or effect that has the greatest impacts Can be repeated in order to improve analysis example in one chart minor categories may be included as major categories in another chart Disadvantages: Can only be used for discrete data type Not appropriate for many data categories Example: Question 3 Mean mode and median: We explain these values using the data set 1, 2, 2, 3, 4, 5, 5 and 5 Mean: The mean, mode and median are all measures of central tendencies of data; the mean is determined by adding up all the values in a data set and then dividing the number of observations. In the example the value of the mean will 27/8 = 3.375 Advantage: It is mostly used especially when comparing two data sets It is unique in that a data set will only have a certain mean value Disadvantages: It is affected by outliers Mode: The mode is the value that appears most times in a data set, for the example 5 appears most times and therefore the mode of the data set is 5. Advantage: It is not affected by outliers Disadvantages: Not unique given that a data set may have more than one mode value When a data set has equal occurrence of all the variables then all the observations are the mode of the data set If a data set has more than one mode then it is difficult to compare it with other data sets. Less often used compared to the mean and median Median: The Median is the middle value of an ordered data set for odd observations or the mean value of the middle two variables for even observations. In the example the median will be 3 +4 = 7, then 7/2 = 3.5, therefore median = 3.5 Advantage: The median is not affected by outliers It is unique in that a data set will only have a certain median value It is useful when comparing two data sets Disadvantages: Less often used compared to the mean which is widely used Question 4 The empirical rule states that given a large sample then the cluster of data around the mean of the data set will assume a normal distribution where 68% of the data will be contained plus or minus one standard deviation, 95% of the data will be within two standard deviations and 99% of the data will be within three standard deviation. The following diagram summarizes the empirical rule: Question 5 Mutually exclusive and collective exhaustive events: Mutually exclusive events are events that cannot occur together at the same time, collective exhaustive events is a set of events where one of them must occur, for example when tossing a coin it is either a head or a tail and one event must occur and this is a collective exhaustive event. Question 6 Statistical independence and conditional probability: Statistical independence in probability with reference to probability states that for two events A and B if P (A B) = P (A). P (B). this concept is related to conditional probability which states that for two independent events A and B the conditional probability of A occurring given B which is stated as P(A/B) = P(A B)/ P(B) Question 7 Properties of a normal distribution: The normal distribution curve is bell shaped and symmetric, it has a single peak and they satisfy the empirical rule that states that 68% of the data will be contained plus or minus one standard deviation, 95% of the data will be within two standard deviations and 99% of the data will be within three standard deviations. Therefore the properties include: -Symmetric -Bell shaped -Single peak. Question 8 Determining whether a data set is normally distributed: In order to determine whether a data is set is normally distributed the skewness value is determined, positive or negative skewness value shows that the data is not normally distributed. Another way to determine is using the values of the mean, median and mode, when these values are all equal meaning that the mean = mode = median then the data set is normally distributed. Question 9 Why large sample follow normal distribution: The central limit theorem states that given a large number of a random variable the distribution is normally distributed as the number of these variables increase indefinitely. If a sample is not normally distributed as we add more observations the observations tend to be concentrated near the mean and this is refereed to as convergence of the limit and therefore the large sample assumes a normal distribution. Question 10 Confidence by increasing confidence interval Given a sample size n confidence can be increased by increasing the confidence interval, for example given the estimated mean of a population is 4 Assume that the confidence interval for A level of confidence is 2 to 6, by increasing the confidence interval and the new level is A + X then the new range will be say 1 to 7, therefore the more confident we are given that the range has increased Question 11 Hypothesis testing between two independent mean and two dependent mean: Independent samples comparison refers to a comparison of two samples that are not related while dependent samples refers to related data sets that may be within the subject or related. An example of dependent samples is a study repeated twice. Hypothesis testing on the difference between means differs for independent and dependent samples and the following are the formulas used to determine the test statistics. Independent sample: Ts = (1 -2) - (X1 - X2)/ [(S12/n1) - (S22/n2)] Dependent samples: Ts = D/ [(ND2 - (D) 2 / (N-1)] Where D is the difference between observations Question 12 ANOVA among group and within group variation: Within group variation - this refers to the variation of data within a group, example when comparing two samples A and B, within group variation will be determined using sample An only and then sample B only. Between groups variation- this is variations between two samples, example variations between groups A variables and group B variables. Question 13 Assumptions of ANOVA: The assumptions of ANOVA include: 1. Independence - this assumption states that the two or more samples being tested are independent and therefore not related in any way. 2. Homogeneity - this assumption states that the variance of the two groups being studies are equal and therefore the sample variance should not be heteroskedastic. 3. Normal distribution- this assumption states that the data in each group is normally distributed. Question 14 Null hypothesis: The null hypothesis refers to the hypothesis being tested, it equates the variable to zero, this is the hypothesis that is tested and not the alternative hypothesis, it is stated as H0, for a simple hypothesis we may state the null hypothesis as: H0: B = 0 This hypothesis states that B = 0, by accepting this hypothesis when the critical value is greater than the calculated statistics it means that we have accepted B = 0 at that level of test. If we reject the null hypothesis this means that we accept the alternative hypothesis that may state that B0. Reference: Anderson, R. (2007) Statistics for Business and Economics. McGraw Hill Press, New York Read More