QMB 3200 Final Exam | Complete Solutions (Answered) Dataset all the data collected for a particular analysis Data The facts & figures collected, analyzed, and summarized for presentation and interpretation. Element Th
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QMB 3200 Final Exam | Complete Solutions (Answered) Dataset all the data collected for a particular analysis Data The facts & figures collected, analyzed, and summarized for presentation and interpretation. Element The entity on which data is collected. Variable a characteristic of interest of an element. Observation The variables associated with an individual element. Categorical Data Use numeric or ordinal values of measurement of categories. Quantitative Data use numeric (quantitative) measures Cross-sectional Data data collected at a similar point in time. Time Series data collected over several time periods. Panel Data combination of cross-sectional and time series data. Descriptive Statistics Describes data or variables Population Is the set of all data/variables of a statistical analysis. Sample is a subset of the population Statistical Inference: Uses data from a sample to make estimates and test hypothesis about the characteristics of a population. What does row 1 usually contain in Excel? Typically contains the variable's names. What does Column A usually contain in Excel? contains the elements; and the rest of the worksheet contains the data in the dataset. How do you calculate the mean in excel? =Average How do you calculate the median in excel? =median What is data analytics? the scientific process of transforming data for decision making. There are three broad areas of data analytics. Descriptive Analytics describe what has happened in the past. Predictive Analytics uses statistical models from past data to predict the future [forecasting] or access the impact of one variable on another [inference]. Prescriptive Analytics uses models seeking to find a best (optimal) solution. Often these are sometype of optimization model. The difference between data and big data are We will use data (not big data) 1. Volume - the number of observations. 2. Velocity - the speed at which data is collected. 3. Variety - the forms of data are of different types. 4. Veracity - the reliability of the data generated. Data Mining Focuses on extracting predictive information from big data. Frequency Distribution a tabular summary of data showing the number (i.e. frequency) of observations in each of several non over-lapping categories. Relative Frequency frequency of a class/ n of a class Percent Frequency relative frequency x 100 Bar Chart: a visual display of frequency; relative frequency & percent frequency distributions. How would you make a frequency table in excel? a. Select any cell in Column A b. Click Insert on the Ribbon Tab c. In Tables, click Recommend Pivot Tables d. Click OK. Pie chart a visual display of frequency; relative frequency & percent frequency distributions. How would you create a bar chart in excel? a. Select any cell from A1 to A51. b. Click the Insert tab from the Ribbon. c. In Charts, click Recommended Charts d. Click OK. [bar chart appears in a new worksheet] What are the basic steps to using excel? Access the data, Functions & formulas, Apply tools, Editing options A frequency distribution with quantitative data must define the classes for a frequency distribution by: a. determine the number of non over-lapping classes; b. determine the width of each class; c. determine the class limits. Number of Classes Typically, between 5 and 20. Small datasets have less; larger datasets have more. Width of the Class Generally, it should be the same for each class. Approximate class width = (largest data value - smallest data value)/number of classes. Class Limits Each data observation must only belong to one class. Relative Frequency Distributions frequency of the class/n To construct a frequency distribution in excel we must group the rows. What if a class has no data? [Excel automatically removes these classes] To change this: Click the Layout and Print tab Choose show items with no data; click OK Histogram A visual display of a frequency, relative frequency, or percent frequency distribution, where the variable of interest is on the horizontal axis and the frequency, relative frequency, or percent frequency is on the vertical axis. A histogram shows shows the shape of the distribution of the variable of interest. A distribution is skewed if more of the data is either to the left or right of the distribution. How would we construct a histogram in excel? a. Select any cell in the pivot table report (A3:B9) b. Click Insert on the Ribbon. c. In the Charts group, click Recommended Charts d. Click OK. set the gap width to 0 Cumulative Distributions Presents the number of data items with values less than or equal to the upper-class limit for each class Cumulative relative frequency distribution A cumulative relative frequency distribution shows the proportion of data items with values less than or equal to the upper limit of each class. Cumulative percent frequency distribution A cumulative percent frequency distribution shows the percentage of data items with values less than or equal to the upper limit of each class. Crosstabulation: a tabular summary of data for two variables (either categorical or quantitative) EX: Suppose we have data from a sample of 300 restaurants on overall quality and the meal price. (This allows us to see if there is a pattern the two variables). Scatter Diagram & Trendline: a scatter diagram is a graphical display of the relationship between two quantitative variables and a trendline provides an approximation (i.e. an estimate) of the relationship; which can be positive, negative or none. How would you create a scatter diagram in excel? a. Select cells B1:C11. b. Click Insert on the Ribbon. c. In the Charts group, click Insert Scatter (X,Y) or Bubble chart (or Recommended Chart) d. When the list of scatter subtypes appears, click scatter. e. In chart elements- Select the checkbox for Trendline. Side-by-Side & Stacked Bar Charts These are extensions of a basic bar chart as they are used to display and compare two variables. stacked bar chart A stacked bar chart has one bar broken into segments of a different color showing the relative frequency of each class. Side by side bar chart A side-by-side bar chart depicts multiple bar charts on the same display. How would we construct a side by side bar chart in excel? c. In the Charts group, click recommended charts (preview) d. Click OK Mean (or average): is the average value for a variable and the sample mean is denoted as Μ
π₯, and the population mean is denoted as π. Ξ£ The Sum symbol π₯π (x sub i) a piece of data in the data set EX:12345- 4=xi n and N n is the number of observations in a sample N is the number of observations in a population Median is the value in the middle, when the data are arranged in ascending order. When the number of observations are odd the median is the middle value; when the number of observations are even the median is the average of the two middle values. The median avoid problems where there are extreme high or low values for x. Mode Mode is the value that occurs with the greatest frequency. If there are two values that are most frequent the variable is bi-modal; if there are more then it's multi-modal. How would we calculate mode in excel? =mode.sngl Weighted Mean: used when observations have different weights (relative importance). Geometric Mean: is a measure of location by finding the nth root of the product of n values. How would we calculate geometric mean in excel? =geomean Percentile: provides information about how the data is spread over the interval from the smallest to the largest value. The pth percentile divides the data into two parts - approximately p% of the observations are less than the pth percentile and approx. (100-p)% are greater. Quartiles: represent how the data is spread over four parts, each containing approximately 25% of the observations. Q1 is the first quartile (25th percentile); Q2 = second quartile (50th percentile or median); Q3 = third quartile (75th percentile). How would we calculate percentile in excel? =percentile.exc(B2:B13,k) A percentile calculated with . 4 as k means 40% percent of values are less than or equal to the calculated result How would we calculate quartiles in excel? = quartile.exc(B2:B13,quartile number) You would select either 1, 2,3, or 4 depending on which quartile you would like to know. These are measures of variability or dispersion of the data. Range: largest value and smallest value. Interquartile Range: Q 3 - Q 1 , is the range of the middle 50% of the data. Variance: measures variability using all the data, based on the difference between the value of xi and the mean. This difference is called a deviation from the mean. Variance: The population variance: π^ 2 The sample variance: π ^2 The benefit is that it is useful in comparing the variability of two or more variables; but one difficulty is the units, since the variance is in units squared. Standard Deviation: The population standard deviation: π= Sqrt. of π^2 The sample standard deviation: π = Sqrt. of π ^2 The advantage is now the units are no longer squared, making it easier to compare the results to the mean as well as other statistics. How would we calculate Variance for a sample in excel? =var.s( How would we calculate standard deviation for a sample in excel? =stdev.s( Coefficient of Variation This is a measure of how large the standard deviation is relative to the mean. How to use descriptive statistics in excel: a. Click on Data in the Ribbon. b. Under Analysis, click on Data Analysis. c. Choose Descriptive Statistics. Select the checkbox for Summary Statistics Click OK. Distribution Shape is measured by skewness. If the shape of the data is skewed to the left, the skewness is negative; if to the right then skewness is positive; and if the data is symmetric, then skewness is zero. For a symmetric distribution, the mean and median are equal. For a positively skewed distribution, the mean is usually greater than the median and a negatively skewed distribution, the mean is usually less than the median. How would we calculate skewness in excel? =SKEW Z scores The z-Score yields a standardized value and is the number of standard deviations from the mean. The z-Score for any observation is a measure of the relative location of the observation in the dataset. Measures the relative location of values in the dataset. This helps determine how far a particular value is from the mean. z-Scores use the mean and standard deviation in its calculation. Chebyshev's Theorem Allows us to make statements about the population of the data values that must be within a specified number of standard deviations from the mean. At least (1 β1/ π§^2) of the data values must be within z standard deviations of the mean (z > 1). It applies to any dataset. If the data is bell shaped around the mean, we know Approx. 68% of the data is within one s of Μ
π₯. Approx. 95% of the data is within two s of Μ
π₯. Approx. 99.7% of the data is within three s of Μ
π₯. Outliers are extreme values relative to the rest of the data. z-Score can help identify outliers. Typically, any z-Score greater than 3 is an outlier. Alternatively, we can use the interquartile range, where the: lower limit: π1 β 1.5(πΌπR) upper limit: π3 + 1.5(πΌQr) Covariance is a descriptive measure of the linear association between two variables. If Covariance or s sub-xy> 0, then there is a positive linear association between x and y. If sxy < 0, then there is a negative linear association between x and y. Be careful about attributing the size of the covariance and the strength of the linear relationship because covariance depends on the units of measurement. Correlation coefficient The sample correlation coefficient ranges from -1 to +1. If it is exactly 1 (perfect positive linear relationship), then all the data would be on a positively sloped line; and if it is exactly -1 (perfect negative linear relationship), then all the data would be on a negatively sloped line. As the data deviates from the slope of the line, the correlation coefficient moves closer to zero. How would we calculate covariance of a sample in excel? =covariance.s How would we calculate the Correlation Coefficient in excel? =correl( Probability A numerical measure of the likelihood of an event occurring. A probability ranges from 0 to 1, such as the probability it will rain tomorrow. Experiment Is a process generating well-defined outcomes. Rolling a 6-sided die results in six possible outcomes: S = {1, 2, 3, 4, 5, 6} Counting rules for experiments If an experiment has k steps with n1 possible outcomes on step 1, n 2 possible outcomes on step 2, and so on, then the total number of experimental outcomes is (n1 )(n2 )...(nk ). Thus, tossing a coin 6 times yields: (2)(2)(2)(2)(2)(2) =64. Permutations a counting rule computing the number of experimental outcomes when n objects are to be selected from a set of N objects where the order of selections is important. Assigning Probabilities requirements 1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively 2. The sum of the probabilities for all experimental outcomes must be equal to 1. Three methods to assign probability: 1. Classical Method - such as a coin toss or a roll of a 6-sided die. 2. Relative Frequency Method: is used when data are available to estimate the proportion of time the experimental outcome will occur if the experiment is repeated a large number of times. 3. Subjective Method: is used when outcomes are not equally likely and data is unavailable. Events are a collection of sample points. Example: Project Completion in 10 months or less: C = {(2,6);(2,7);(2,8);(3,6);(3,7);(4,6)} Probability of an event The probability of an event is equal to the sum of the probabilities of the sample points in the event. Thus, P(C) = P(2,6)+P(2,7)+P(2,8)+P(3,6)+P(3,7)+P(4,6) P(C) = 0.15 + 0.15 + 0.05 + 0.10 + 0.20 + 0.05 = 0.70 Complement of Event are all the sample points not in the event. Union of Two Events the event containing all sample points belonging to Event A, Event B, or both. The union of Event A and Event B is denoted by: π΄ βͺ π΅. Intersection of Two Events is the event containing the sample points belonging to only both A and B. The intersection is denoted by: π΄β©π΅. Addition Law is useful when we want to know the probability that at least one of two events occurs. π(π΄ βͺ π΅) = π(π΄) + π(π΅) β π(π΄ β© π΅). Mutually Exclusive Events occur when two events have no sample points in common. Addition Law for mutually exclusive events: π(π΄ βͺ π΅) = P (A) + P(B) Conditional Probability Often the probability of an event is influenced by whether a related event already occurred. Suppose Event A occurs with P(A). If Event B already occurred, this new information results in a new probability for Event A, and is called a conditional probability: P(A|B) Joint Probability The probability of the intersection of two events. Marginal Probabilities is the sum of the joint probabilities (by row and column). Conditional Probability equations π(π΄Iπ΅) = π(π΄ β© π΅)/π(π΅) or π(π΅Iπ΄) = π(π΄ β© π΅)/π(π΄) Independent Events Event A and Event B are independent if: P(A|B) = P(A) or P(B|A) =P(B) Multiplication Law is used to compute the probability of the intersection of two events π (π΄ β© π΅) = π(π΅) β π(π΄|π΅) For independent events: π( π΄ β© π΅) = π(π΄) β π(π΅) Random Variable A numeric description of the outcome of an experiment and is either discrete or continuous. Discrete Random Variables either are a finite number of values or an infinite number of values such as 0, 1,2 ... Examples are a toss of a coin, the number of customers who place an order or the product chosen by a customer from two options. Continuous Random Variables are any numerical value in an interval or collection of intervals. Examples are the time a customer visits a webpage (xβ₯ 0); ounces in a soft drink can (0 β€ x β€ 12.1oz) Probability distribution of a random variable describes how probabilities are distributed over the values of the random variable. For a discrete random variable (x), a probability function f(x) provides the probability for each value of the random variable.Note: f(x) β₯ 0 and Ξ£ f(x) = 1 Discrete probability example x = number of cars sold; f(x) is the probability distribution as a relative frequency. Thus, the probability of selling 3 or more cars is: f(x β₯ 3) = f(3) + f(4) + f(5) = 0.14 + 0.04 + 0.01 = 0.19 For discrete random variables, we compute the probability at a specific value of x. Expected value or mean of a random variable Expected Value (or mean) of a random variable is a measure of the central location for a random variable. E(x) = ΞΌ = Ξ£ x*f(x), Standard Deviation is the positive square root of the variance. Excel functions for a probability distribution Mean = sumproduct(A:A,B:B) Squared Deviation from Mean: =(A2-F$2)^2 - (the data number minus the mean) squared Variance = sumproduct(C:C,B:B) - the sum product of the probability and the mean. Standard Deviation = sqrt(B11) How do you keep columns and rows from changing in excel? Put a $ before the column and row to keep them from changing. Bivariate Distribution A bivariate probability distribution involves two random variables, such as rolling a die two times or recording the percentage change for a stock fund and a bond fund over a year. Often the analyst is interested in the relationship between the two random variables, we look at the covariance and correlation coefficient as measures of the linear association between the random variables. What is one way to measure financial risk of an asset? Standard deviation Binomial Probability Distribution Is based on the following four properties: 1. The experiment consists of a sequence of n identical trials. 2. Two outcomes are possible on each trial; success or failure. 3. The probability of success (p) and the probability of failure (1-p) does not change from trial to trial. 4. The trials are independent. We are interested in the number of successes in the n trials. Since x has values 0, 1, 2, ... n, x is finite and a discrete random variable, and the probability distribution from this random variable is the binomial probability distribution. A classic example is consecutive coin tosses - which meets the four properties. How would we calculate a binomial probability for excel? =Binom.Dist(x,trials,probability of success,True/FALSE) Binom.Dist first needs a value for x, then a value for n, a value of p, and finally either TRUE (cumulative probability) or FALSE (probability). Poisson Probability Distribution This distribution relates to the case for estimating the number of occurrences over a specified interval of space/time. The mean and the variance are equal. Ex: Number of patients arriving at an emergency room during a 15-minute weekday morning interval Properties of a Poisson Experiment 1. The probability of an occurrence is the same for any two intervals of equal length. 2. Occurrence or non-occurrence in any interval is independent of the occurrence or non-occurrence in any other interval. How would we calculate a poisson distribution in excel? =Poisson.Dist(x,ΞΌ,True/FALSE) Use Poisson.Dist, where we need the value of x, the value of ΞΌ and TRUE (cumulative probability) or FALSE (probability). Hypergeometric Probability Distribution The Hypergeometric Probability Distribution is similar to the binomial distribution, except the trials are not independent and the probability of success changes from trial to trial. NOTE: r is the number of successes in population N, and N-r is the number of failures. EX: We are taking out 5 items from a toy line and need to know the probability of the toys being defective or working. How would we calculate hypergeometric probability distribution in excel? =Hypergeom.Dist(x,ΞΌ,r,N,TRUE/FALSE) Hypergeom.Dist needs a value for x, a value for ΞΌ, a value of r, a value of N, and either TRUE or FALSE. To drag data down in excel you can also double click on the box and it will autofill the rest of the data in the column. To change a probability distribution in excel from regular to cumulative what would you change? You would change from false to true to change to cumulative. Continuous Random Variable For continuous random variables we compute the probability that the random variable assumes any value in an interval. Thus, we are computing the area under the probability density function, f(x). EX:Suppose a flight time (x) is anywhere between 120 to 140 minutes. Because x can take any value between 120 and 140, x is a continuous random variable. Computed differently than a descrete random variable. Uniform Probability Distribution If historic flight data shows the probability of flight time in 1-minute intervals is equally the same as any other 1-minute interval, then x has a uniform probability distribution. The uniform probability distribution is a rectangle; thus, the area is length times height. The uniform probability distribution has a total area equal to 1, which is true of all continuous probability density functions. Shown as P(120 β€ x β€ 130)
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