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Based on your Module topics, what did you find new and interesting?  And what appeared to be a review?  Also, identify at least one discussion post you found interesting, helpful, or beneficial (and why).    Topics covered in this Module Data visualization Table design principles Chart types PivotCharts Advanced data visualization Data dashboards Events and probabilities Conditional probability Random variables Discrete probability distributions Continuous probability distributions Learning objectives By the end of this module, students should be able to: Explain design techniques for data visualization Create pivot tables, scatter charts, bar charts, bubble charts, pivotcharts in Excel Identify applications for data dashboards Describe events and probabilities Explain conditional probability, Bayes' Theorem, Multiplication Law Discuss random variables and probability distribution Calculate in Excel expected value variance standard deviation binomial probabilities poisson probabilities mean exponential probabilities Business Analytics © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Data Visualization Chapter 3 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction Data visualization involves: Creating a summary table for the data. Generating charts to help interpret, analyze, and learn from the data. Uses of data visualization: Helpful for identifying data errors. Reduces the size of your data set by highlighting important relationships and trends in the data. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 3 Overview of Data Visualization Effective Design Techniques © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Overview of Data Visualization (Slide 1 of 5) Effective Design Techniques: Data-ink ratio: Measures the proportion of what Tufte terms “data-ink” to the total amount of ink used in a table or chart. Edward R. Tufte first described the data-ink ratio. Helpful for creating effective tables and charts for data visualization: Data-ink: Ink used in a table or chart that is necessary to convey the meaning of the data to the audience. Non-data-ink: Ink used in a table or chart that serves no useful purpose in conveying the data to the audience. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Overview of Data Visualization (Slide 2 of 5) Table 3.1: Example of a Low Data-Ink Ratio Table Scarf Sales Day Sales Day Sales 1 150 11 170 2 170 12 160 3 140 13 290 4 150 14 200 5 180 15 210 6 180 16 110 7 210 17 90 8 230 18 140 9 140 19 150 10 200 20 230 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Let us consider the case of Gossamer Industries, a firm that produces fine silk clothing products. Gossamer is interested in tracking the sales of one of its most popular items, a particular style of women’s scarf. Table 3.1 and Figure 3.3 provide examples of a table and chart with low data-ink ratios used to display sales of this style of women’s scarf. The data used in this table and figure represent product sales by day. 6 Overview of Data Visualization (Slide 3 of 5) Figure 3.3: Example of a Low Data-Ink Ratio Chart © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Let us consider the case of Gossamer Industries, a firm that produces fine silk clothing products. Gossamer is interested in tracking the sales of one of its most popular items, a particular style of women’s scarf. Table 3.1 and Figure 3.3 provide examples of a table and chart with low data-ink ratios used to display sales of this style of women’s scarf. The data used in this table and figure represent product sales by day. 7 Overview of Data Visualization (Slide 4 of 5) Table 3.2: Increasing the Data-Ink Ratio by Removing Unnecessary Gridlines Scarf Sales Day Sales Day Sales 1 150 11 170 2 170 12 160 3 140 13 290 4 150 14 200 5 180 15 210 6 180 16 110 7 210 17 90 8 230 18 140 9 140 19 150 10 200 20 230 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 3.2 shows a modified table in which all grid lines have been deleted except for those around the title of the table. Deleting the grid lines in Table 3.1 increases the data-ink ratio because a larger proportion of the ink used in the table is used to convey the information (the actual numbers). Similarly, deleting the unnecessary horizontal lines in Figure 3.4 increases the data-ink ratio. Removing the unnecessary lines makes it easier to read Table 3.2 and Figure 3.4. 8 Overview of Data Visualization (Slide 5 of 5) Figure 3.4: Increasing the Data-Ink Ratio by Adding Labels to Axes and Removing Unnecessary Lines and Labels © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 3.2 shows a modified table in which all grid lines have been deleted except for those around the title of the table. Deleting the grid lines in Table 3.1 increases the data-ink ratio because a larger proportion of the ink used in the table is used to convey the information (the actual numbers). Similarly, deleting the unnecessary horizontal lines in Figure 3.4 increases the data-ink ratio. Removing the unnecessary lines makes it easier to read Table 3.2 and Figure 3.4. 9 Tables Table Design Principles Crosstabulation PivotTables in Excel Recommended PivotTables in Excel © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Tables (1 of 18) Tables should be used when: The reader needs to refer to specific numerical values. The reader needs to make precise comparisons between different values and not just relative comparisons. The values being displayed have different units or very different magnitudes. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 11 Tables (2 of 18) Table 3.3: Table Showing Exact Values for Costs and Revenues by Month for Gossamer Industries Month 1 2 3 4 5 6 Total Costs ($) 48,123 56,458 64,125 52,158 54,718 50,985 326,567 Revenues ($) 64,124 66,128 67,125 48,178 51,785 55,687 353,027 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Consider when the accounting department of Gossamer Industries is summarizing the company’s annual data for completion of its federal tax forms. In this case, the specific numbers corresponding to revenues and expenses are important and not just the relative values. Therefore, these data should be presented in a table similar to Table 3.3. Similarly, if it is important to know exactly by how much revenues exceed expenses each month, then this would also be better presented as a table rather than as a line chart, as seen in Figure 3.5. 12 Tables (3 of 18) Figure 3.5: Line Chart of Monthly Costs and Revenues at Gossamer Industries © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Consider when the accounting department of Gossamer Industries is summarizing the company’s annual data for completion of its federal tax forms. In this case, the specific numbers corresponding to revenues and expenses are important and not just the relative values. Therefore, these data should be presented in a table similar to Table 3.3. Similarly, if it is important to know exactly by how much revenues exceed expenses each month, then this would also be better presented as a table rather than as a line chart, as seen in Figure 3.5. 13 Tables (4 of 18) Figure 3.6: Combined Line Chart and Table for Monthly Costs and Revenues at Gossamer Industries © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Figure 3.6 allows the reader to easily see the monthly changes in revenues and costs while also being able to refer to the exact numerical values. 14 Tables (5 of 18) Table 3.4: Table Displaying Head Count, Costs, and Revenues at Gossamer Industries Month 1 Month 2 Month 3 Month 4 Month 5 Month 6 Total Head Count 8 9 10 9 9 9 Costs ($) 48,123 56,458 64,125 52,158 54,718 50,985 326,567 Revenues ($) 64,124 66,128 67,125 48,178 51,785 55,687 353,027 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Costs and revenues are measured in dollars ($), but head count is measured in number of employees. Although all these values can be displayed on a line chart using multiple vertical axes, this is generally not recommended. Because the values have widely different magnitudes (costs and revenues are in the tens of thousands, whereas headcount is approximately 10 each month), it would be difficult to interpret changes on a single chart. Therefore, a table similar to Table 3.4 is recommended. 15 Tables (6 of 18) Table Design Principles: Avoid using vertical lines in a table unless they are necessary for clarity. Horizontal lines are generally necessary only for separating column titles from data values or when indicating that a calculation has taken place. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 16 Tables (7 of 18) Figure 3.7: Comparing Different Table Designs © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Figure 3.7 compares several forms of a table displaying Gossamer’s costs and revenue data. Most people find Design D, with the fewest grid lines, easiest to read. In this table, grid lines are used only to separate the column headings from the data and to indicate that a calculation has occurred to generate the Profits row and the Total column. 17 Tables (8 of 18) Table 3.5: Larger Table Showing Revenues by Location for 12 Months of Data Revenues by Location ($) Month 1 Month 2 Month 3 Month 4 Month 5 Month 6 Temple 8,987 8,595 8,958 6,718 8,066 8,574 Killeen 8,212 9,143 8,714 6,869 8,150 8,891 Waco 11,603 12,063 11,173 9,622 8,912 9,553 Belton 7,671 7,617 7,896 6,899 7,877 6,621 Granger 7,642 7,744 7,836 5,833 6,002 6,728 Harker Heights 5,257 5,326 4,998 4,304 4,106 4,980 Gatesville 5,316 5,245 5,056 3,317 3,852 4,026 Lampasas 5,266 5,129 5,022 3,022 3,088 4,289 Academy 4,170 5,266 7,472 1,594 1,732 2,025 Total 64,124 66,128 67,125 48,178 51,785 55,687 Costs ($) 48,123 56,458 64,125 52,158 54,718 50,985 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 3.5 breaks out the revenue data by location for 9 cities and shows 12 months of revenue and cost data. Every other column has been lightly shaded. This helps the reader quickly scan the table to see which values correspond with each month. The horizontal line between the revenue for Academy and the Total row helps the reader differentiate the revenue data for each location and indicates that a calculation has taken place to generate the totals by month. Columns of numerical values in a table should be right-aligned. This makes it easy to see differences in the magnitude of values. If you are showing digits to the right of the decimal point, all values should include the same number of digits to the right of the decimal. It is generally best to left-align text values within a column in a table, as in the Revenues by Location (the first) column of Table 3.5. Column headings should either match the alignment of the data in the columns or be centered over the values, as in Table 3.5. 18 Tables (9 of 18) Table 3.5 (cont.) Revenues by Location ($) Month 7 Month 8 Month 9 Month 10 Month 11 Month 12 Total Temple 8,701 9,490 9,610 9,262 9,875 11,058 107,895 Killeen 8,766 9,193 9,603 10,374 10,456 10,982 109,353 Waco 11,943 12,947 12,925 14,050 14,300 13,877 142,967 Belton 7,765 7,720 7,824 7,938 7,943 7,047 90,819 Granger 7,848 7,717 7,646 7,620 7,728 8,013 88,357 Harker Heights 5,084 5,061 5,186 5,179 4,955 5,326 59,763 Gatesville 5,135 5,132 5,052 5,271 5,304 5,154 57,859 Lampasas 5,110 5,073 4,978 5,343 4,984 5,315 56,620 Academy 8,772 1,956 3,304 3,090 3,579 2,487 45,446 Total 69,125 64,288 66,128 68,128 69,125 69,258 759,079 Costs ($) 57,898 62,050 65,215 61,819 67,828 69,558 710,935 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 3.5 breaks out the revenue data by location for 9 cities and shows 12 months of revenue and cost data. Every other column has been lightly shaded. This helps the reader quickly scan the table to see which values correspond with each month. The horizontal line between the revenue for Academy and the Total row helps the reader differentiate the revenue data for each location and indicates that a calculation has taken place to generate the totals by month. Columns of numerical values in a table should be right-aligned. This makes it easy to see differences in the magnitude of values. If you are showing digits to the right of the decimal point, all values should include the same number of digits to the right of the decimal. It is generally best to left-align text values within a column in a table, as in the Revenues by Location (the first) column of Table 3.5. Column headings should either match the alignment of the data in the columns or be centered over the values, as in Table 3.5. 19 Tables (10 of 18) Crosstabulation: A useful type of table for describing data of two variables. PivotTable: A crosstabulation in Microsoft Excel. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 20 Tables (11 of 18) Table 3.6: Quality Rating and Meal Price for 300 Los Angeles Restaurants Restaurant Quality Rating Meal Price ($) Wait Time (min) 1 Good 18 5 2 Very Good 22 6 3 Good 28 1 4 Excellent 38 74 5 Very Good 33 6 6 Good 28 5 7 Very Good 19 11 8 Very Good 11 9 9 Very Good 23 13 10 Good 13 1 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Data on the quality rating, meal price, and the usual wait time for a table during peak hours were collected for a sample of 300 Los Angeles area restaurants. Table 3.6 shows the data for the first 10 restaurants. Quality ratings - categorical data; Meal prices - quantitative data. 21 Tables (12 of 18) Table 3.7: Crosstabulation of Quality Rating and Meal Price for 300 Los Angeles Restaurants Meal Price Quality Rating $10–19 $20–29 $30–39 $40–49 Total Good 42 40 2 0 84 Very Good 34 64 46 6 150 Excellent 2 14 28 22 66 Total 78 118 76 28 300 The greatest number of restaurants in the sample (64) have a very good rating and a meal price in the $20–29 range. Only two restaurants have an excellent rating and a meal price in the $10–19 range. The right and bottom margins of the crosstabulation give the frequencies of quality rating and meal price separately. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. A crosstabulation of the data for quality rating and meal price data is shown in Table 3.7. The left and top margin labels define the classes for the two variables. In the left margin, the row labels (Good, Very Good, and Excellent) correspond to the three classes of the quality rating variable. In the top margin, the column labels ($10–19, $20–29, $30–39, and $40–49) correspond to the four classes (or bins) of the meal price variable. Each restaurant in the sample provides a quality rating and a meal price. For example, restaurant 5 is identified as having a very good quality rating and a meal price of $33. This restaurant belongs to the cell in row 2 and column 3. From the right margin, we see that data on quality ratings show 84 good restaurants, 150 very good restaurants, and 66 excellent restaurants. Similarly, the bottom margin shows the counts for the meal price variable. The value of 300 in the bottom right corner of the table indicates that 300 restaurants were included in this data set. In constructing a crosstabulation, we simply count the number of restaurants that belong to each of the cells in the crosstabulation. 22 Tables (13 of 18) Figure 3.8: Excel Worksheet Containing Restaurant Data © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Figure 3.8 illustrates Zagat’s restaurant data in Excel. Each of the four columns in Figure 3.8 [Restaurant, Quality Rating, Meal Price ($), and Wait Time (min)] is considered a field by Excel. 23 Tables (14 of 18) Figure 3.9: Initial PivotTable Field List and PivotTable Field Report for the Restaurant Data © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. To create a PivotTable in Excel, we follow these steps: Step 1: Click the Insert tab on the Ribbon Step 2: Click PivotTable in the Tables group Step 3: When the Create PivotTable dialog box appears: Choose Select a table or range Enter A1:D301 in the Table/Range: box Select New Worksheet as the location for the PivotTable Report Click OK The resulting initial PivotTable Field List and PivotTable Report are shown in Figure 3.9. 24 Tables (15 of 18) Figure 3.10: Completed PivotTable Field List and a Portion of the PivotTable Report for the Restaurant Data (Columns H:AK Are Hidden) © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Fields may be chosen to represent rows, columns, or values in the body of the PivotTable Report. The following steps show how to use Excel’s PivotTable Field List to assign the Quality Rating field to the rows, the Meal Price ($) field to the columns, and the Restaurant field to the body of the PivotTable report. Step 4: In the PivotTable Fields task pane, go to Drag fields between areas below: Drag the Quality Rating field to the ROWS area Drag the Meal Price ($) field to the COLUMNS area Drag the Restaurant field to the VALUES area Step 5: Click on Sum of Restaurant in the VALUES area Step 6: Select Value Field Settings from the list of options Step 7: When the Value Field Settings dialog box appears: Under Summarize value field by, select Count Click OK Figure 3.10 shows the completed PivotTable Field List and a portion of the PivotTable worksheet as it now appears. 25 Tables (16 of 18) Figure 3.11: Final PivotTable Report for the Restaurant Data © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. To complete the PivotTable, we need to group the columns representing meal prices and place the row labels for quality rating in the proper order: Step 8: Right-click in cell B4 or any cell containing a meal price column label Step 9: Select Group from the list of options Step 10: When the Grouping dialog box appears: Enter 10 in the Starting at: box Enter 49 in the Ending at: box Enter 10 in the By: box Click OK Step 11: Right-click on “Excellent” in cell A5 Step 12: Select Move and click Move “Excellent” to End The final PivotTable, shown in Figure 3.11, provides the same information as the crosstabulation in Table 3.7. For instance, row 8 provides the frequency distribution for the data over the quantitative variable of meal price. A total of 78 restaurants have meal prices of $10 to $19. Column F provides the frequency distribution for the data over the categorical variable of quality. A total of 150 restaurants have a quality rating of Very Good. 26 Tables (17 of 18) Figure 3.12: Percent Frequency Distribution as a PivotTable for the Restaurant Data © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Percent frequency distribution for the restaurant data can be created using a PivotTable by using the following steps: Step 1: To invoke the PivotTable Fields task pane, select any cell in the pivot table Step 2: In the PivotTable Fields task pane, click the Count of Restaurant in the VALUES area Step 3: Select Value Field Settings… from the list of options Step 4: When the Value Field Settings dialog box appears, click the tab for Show Values As Step 5: In the Show values as area, select % of Grand Total from the drop-down menu Click OK Figure 3.12 displays the percent frequency distribution for the Restaurant data as a Pivot-Table. The figure indicates that 50% of the restaurants are in the Very Good quality category and that 26% have meal prices between $10 and $19. 27 Tables (18 of 18) Figure 3.13: PivotTable Report for the Restaurant Data with Average Wait Times Added © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. PivotTables may be used to display statistics other than a simple count of items. The PivotTable in Figure 3.11 can be modified to display summary information on wait times instead of meal prices. Step 1: To invoke the PivotTable Fields task pane, select any cell in the pivot table Step 2: Click the Count of Restaurant field in the VALUES area Select Remove Field Step 3: Drag the Wait Time (min) to the VALUES area Step 4: Click on Sum of Wait Time (min) in the VALUES area Step 5: Select Value Field Settings… from the list of options Step 6: When the Value Field Settings dialog box appears: Under Summarize value field by, select Average Click Number Format In the Category: area, select Number Enter 1 for Decimal places: Click OK When the Value Field Settings dialog box reappears, click OK The completed PivotTable appears in Figure 3.13. This PivotTable replaces the counts of restaurants with values for the average wait time for a table at a restaurant for each grouping of meal prices ($10–19, $20–29, $30–39, $40–49). For instance, cell B7 indicates that the average wait time for a table at an Excellent restaurant with a meal price of $10–$19 is 25.5 minutes. Column F displays the total average wait times for tables in each quality rating category. We see that Excellent restaurants have the longest average waits of 35.2 minutes and that Good restaurants have average wait times of only 2.5 minutes. Finally, cell D7 shows us that the longest wait times can be expected at Excellent restaurants with meal prices in the $30–$39 range (34 minutes). 28 Charts Scatter Charts Recommended Charts in Excel Line Charts Bar Charts and Column Charts A Note on Pie Charts and Three-Dimensional Charts Bubble Charts Heat Maps Additional Charts for Multiple Variables PivotCharts in Excel © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Charts (1 of 26) Charts (or graphs): Visual methods of displaying data. Scatter chart: Graphical presentation of the relationship between two quantitative variables. Trendline: A line that provides an approximation of the relationship between the variables. Line chart: A line connects the points in the chart. Useful for time series data collected over a period of time (minutes, hours, days, years, etc.). © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30 Charts (2 of 26) Table 3.8: Sample Data for the San Francisco Electronics Store No. of Commercials Sales ($100s) Week x y 1 2 50 2 5 57 3 1 41 4 3 54 5 4 54 6 1 38 7 5 63 8 3 48 9 4 59 10 2 46 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. We will investigate whether a relationship exists between the number of commercials shown and sales at the store the following week using a scatter chart. The steps involved to create a scatter chart using Excel’s chart tools are as follows: Copy the data in the file Electronics to a new excel worksheet in columns A through C and rows 1 through 11. Step 1: Select cells B2:C11 Step 2: Click the Insert tab in the Ribbon Step 3: Click the Insert Scatter (X,Y) or Bubble Chart button in the Charts group Step 4: When the list of scatter chart subtypes appears, click the Scatter button Step 5: Click the Design tab under the Chart Tools Ribbon Step 6: Click Add Chart Element in the Chart Layouts group Select Chart Title, and click Above Chart Click on the text box above the chart, and replace the text with Scatter Chart for the San Francisco Electronics Store Step 7: Click Add Chart Element in the Chart Layouts group Select Axis Title, and click Primary Horizontal Click on the text box under the horizontal axis, and replace “Axis Title” with Number of Commercials Step 8: Click Add Chart Element in the Chart Layouts group Select Axis Title, and click Primary Vertical Click on the text box next to the vertical axis, and replace “Axis Title” with Sales ($100s) Step 9: Right-click on the one of the horizontal grid lines in the body of the chart, and click Delete Step 10: Right-click on the one of the vertical grid lines in the body of the chart, and click Delete To add a linear trendline using Excel, we use the following steps: Step 1: Right-click on one of the data points in the scatter chart, and select Add Trendline… Step 2: When the Format Trendline task pane appears, select Linear under Trendline Options 31 Charts (3 of 26) Figure 3.17: Scatter Chart for the San Francisco Electronics Store © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The number of commercials (x) is shown on the horizontal axis, and sales (y) are shown on the vertical axis. For week 1, x = 2 and y = 50. A point is plotted on the scatter chart at those coordinates; similar points are plotted for the other nine weeks. Note that during two of the weeks, one commercial was shown, during two of the weeks, two commercials were shown, and so on. The completed scatter chart in Figure 3.17 indicates a positive linear relationship (or positive correlation) between the number of commercials and sales: higher sales are associated with a higher number of commercials. 32 Charts (4 of 26) Table 3.9: Monthly Sales Data of Air Compressors at Kirkland Industries Month Sales ($100s) Jan 135 Feb 145 Mar 175 Apr 180 May 160 Jun 135 Jul 210 Aug 175 Sep 160 Oct 120 Nov 115 Dec 120 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Table 3.9 contains total sales amounts (in $100s) for air compressors during each month in the most recent calendar year. To create the line chart to this data, the steps are as follows: Copy the data in the file Kirkland to a new Excel worksheet in columns A and B; rows 1 through 13. Step 1: Select cells A2:B13 Step 2: Click the Insert tab on the Ribbon Step 3: Click the Insert Line Chart button in the Charts group Step 4: When the list of line chart subtypes appears, click the Line with Markers button under 2-D Line This creates a line chart for sales with a basic layout and minimum formatting Step 5: Select the line chart that was just created to reveal the Chart Buttons Step 6: Click the Chart Elements button Select the check boxes for Axes, Axis Titles, and Chart Title. Deselect the check box for Gridlines. Click on the text box next to the vertical axis, and replace “Axis Title” with Sales ($100s) Click on the text box next to the horizontal axis and replace “Axis Title” with Month. Click on the text box above the chart, and replace “Sales ($100s)” with Line Chart for Monthly Sales Data 33 Charts (5 of 26) Figure 3.19: Scatter Chart and Line Chart for Monthly Sales Data at Kirkland Industries © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 34 Charts (6 of 26) Table 3.10: Regional Sales Data by Month for Air Compressors at Kirkland Industries … Business Analytics © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Probability: An Introduction to Modeling Uncertainty Chapter 4 © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2 Introduction Uncertainty is an ever-present fact of life for decision makers. Much time and effort are spent trying to plan for and respond to uncertainty. Probability is the numerical measure of the likelihood that an event will occur. This measure of uncertainty is often communicated through a probability distribution: Extremely helpful in providing additional information about an event. Can be used to help a decision maker evaluate possible actions and determine best course of action. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 3 Events and Probabilities © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 4 Events and Probabilities (Slide 1 of 6) A random experiment is a process that generates well-defined outcomes. By specifying all possible outcomes, we identify the sample space for a random experiment; examples: A coin toss. Rolling a die. An event is defined as a collection of outcomes. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 5 Events and Probabilities (Slide 2 of 6) Table 4.1: Random Experiments and Experimental Outcomes Random Experiment Experimental Outcomes Toss a coin Head, tail Roll a die 1, 2, 3, 4, 5, 6 Conduct a sales call Purchase, no purchase Hold a particular share of stock for one year Price of stock goes up, price of stock goes down, no change in stock price Reduce price of product Demand goes up, demand goes down, no change in demand © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 6 Events and Probabilities (Slide 3 of 6) Example: California Power & Light Company (CP&L). CP&L is starting a project designed to increase the generating capacity of one of its plants in southern California. Analysis of similar construction projects indicates that the possible completion times for the project are 8, 9, 10, 11, and 12 months. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 7 Events and Probabilities (Slide 4 of 6) Table 4.2: Completion Times for 40 CP&L Projects © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 8 Events and Probabilities (Slide 5 of 6) The probability of an event is equal to the sum of probabilities of outcomes for the event. CP&L example: Let C denote the event that the project is completed in 10 months or less, C = {8,9,10}. The probability of event C, denoted We can tell CP&L management that there is a 0.70 probability that the project will be completed in 10 months or less. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 9 Some Basic Relationships of Probability Complement of an Event Addition Law © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 10 Some Basic Relationships of Probability (Slide 1 of 11) Completion of an Event: Given an event A, the complement of A is defined to be the event consisting of all outcomes that are not in A. Figure 5.1 shows what is known as a Venn diagram, which illustrates the concept of a complement: Rectangular area represents the sample space for the random experiment and contains all possible outcomes. Circle represents event A and contains only the outcomes that belong to A. Shaded region of the rectangle contains all outcomes not in event A. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 11 Some Basic Relationships of Probability (Slide 2 of 11) Figure 4.1: Venn Diagram for Event A © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 12 Some Basic Relationships of Probability (Slide 3 of 11) The probability of an event A can be computed easily if the probability of its complement is known. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 13 Some Basic Relationships of Probability (Slide 4 of 11) Addition Law: The addition law is helpful when we are interested in knowing the probability that at least one of two events will occur. Concepts related to the combination of events: The union of events. The intersection of events. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 14 Some Basic Relationships of Probability (Slide 5 of 11) Given two events A and B, the union of A and B is defined as the event containing all outcomes belonging to A or B or both. The union of A and B is denoted by The Venn diagram in Figure 4.2 depicts the union of A and B: One circle contains all the outcomes of A. The other circle contains all the outcomes of B. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 15 Some Basic Relationships of Probability (Slide 6 of 11) Figure 4.2: Venn Diagram for the Union of Events A and B © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 16 Some Basic Relationships of Probability (Slide 7 of 11) The definition of the intersection of A and B is the event containing the outcomes that belong to both A and B. The intersection of A and B is denoted by The Venn diagram depicting the intersection of A and B is shown in Figure 4.3: The area in which the two circles overlap is the intersection. It contains outcomes that are in both A and B. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17 Some Basic Relationships of Probability (Slide 8 of 11) Figure 4.3: Venn Diagram for the Intersection of Events A and B © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 18 Some Basic Relationships of Probability (Slide 9 of 11) The addition law provides a way to compute the probability that event A or event B or both will occur. Used to compute the probability of the union of two events. A special case arises for mutually exclusive events: If the occurrence of one event precludes the occurrence of the other. If the events have no outcomes in common. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 19 Some Basic Relationships of Probability (Slide 10 of 11) Figure 4.4: Venn Diagram for Mutually Exclusive Events © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 20 Some Basic Relationships of Probability (Slide 11 of 11) © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 21 Conditional Probability Independent Events Multiplication Law Bayes’ Theorem © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 22 Conditional Probability (Slide 1 of 21) Conditional probability: When the probability of one event is dependent on whether some related event has already occurred. Illustration: Lancaster Savings and Loan: Interested in mortgage default risk. Interested in whether the probability of a customer defaulting differs by marital status. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 23 Conditional Probability (Slide 2 of 21) Table 4.3: Subset of Data from 300 Home Mortgages of Customers at Lancaster Savings and Loan Customer No. Age Marital Status Annual Income Mortgage Amount Payments per Year Total Amount Paid Default on Mortgage? 1 37 Single $172,125.70 $473,402.96 24 $581,885.13 Yes 2 31 Single $108,571.04 $300,468.60 12 $489,320.38 No 3 37 Married $124,136.41 $330,664.24 24 $493,541.93 Yes 4 24 Married $79,614.04 $230,222.94 24 $449,682.09 Yes 5 27 Single $68,087.33 $282,203.53 12 $520,581.82 No 6 30 Married $59,959.80 $251,242.70 24 $356,711.58 Yes © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 24 Conditional Probability (Slide 3 of 21) Table 4.3: Continued Customer No. Age Marital Status Annual Income Mortgage Amount Payments per Year Total Amount Paid Default on Mortgage? 7 41 Single $99,394.05 $282,737.29 12 $524,053.46 No 8 29 Single $38,527.35 $238,125.19 12 $468,595.99 No 9 31 Married $112,078.62 $297,133.24 24 $399,617.40 Yes 10 36 Single $224,899.71 $622,578.74 12 $1,233,002.14 No 11 31 Married $27,945.36 $215,440.31 24 $285,900.10 Yes 12 40 Single $48,929.74 $252,885.10 12 $336,574.63 No 13 39 Married $82,810.92 $183,045.16 12 $262,537.23 No © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 25 Conditional Probability (Slide 4 of 21) Table 4.3: Continued Customer No. Age Marital Status Annual Income Mortgage Amount Payments per Year Total Amount Paid Default on Mortgage? 14 31 Single $68,216.88 $165,309.34 12 $253,633.17 No 15 40 Single $59,141.13 $220,176.18 12 $424,749.80 No 16 45 Married $72,568.89 $233,146.91 12 $356,363.93 No 17 32 Married $101,140.43 $245,360.02 24 $388,429.41 Yes 18 37 Married $124,876.53 $320,401.04 4 $360,783.45 Yes 19 32 Married $133,093.15 $494,395.63 12 $861,874.67 No 20 32 Single $85,268.67 $159,010.33 12 $308,656.11 No © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 26 Conditional Probability (Slide 5 of 21) Table 4.3: Continued Customer No. Age Marital Status Annual Income Mortgage Amount Payments per Year Total Amount Paid Default on Mortgage? 21 37 Single $92,314.96 $249,547.14 24 $342,339.27 Yes 22 29 Married $120,876.13 $308,618.37 12 $472,668.98 No 23 24 Single $86,294.13 $258,321.78 24 $380,347.56 Yes 24 32 Married $216,748.68 $634,609.61 24 $915,640.13 Yes 25 44 Single $46,389,75 $194,770.91 12 $385,288.86 No © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 27 Conditional Probability (Slide 6 of 21) Table 4.4: Crosstabulation of Marital Status and if Customer Defaults on Mortgage Marital Status No Default Default Total Married 64 79 143 Single 116 41 157 Total 180 120 300 From Table 4.4 or Figure 4.5, the probability that a customer defaults on his or her mortgage is The probability that a customer does not default on his or her mortgage is © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 28 Conditional Probability (Slide 7 of 21) Figure 4.5: PivotTable for Marital Status and Whether Customer Defaults on Mortgage © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 29 Conditional Probability (Slide 8 of 21) When values give the probability of the intersection of two events, the probabilities are called joint probabilities. Marginal probabilities are found by summing the joint probabilities in the corresponding row or column of the joint probability table. Conditional probabilities can be computed as the ratio of joint probability to a marginal probability. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30 Conditional Probability (Slide 9 of 21) Table 4.5: Joint Probability Table for Customer Mortgage Prepayments © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 31 Conditional Probability (Slide 10 of 21) Figure 4.6: Using Excel PivotTable to Calculate Conditional Probabilities © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 32 Conditional Probability (Slide 11 of 21) Independent Events: If the probability of event D is not changed by the existence of event M, then we would say that events D and M are independent events. Otherwise, the events are dependent. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 33 Conditional Probability (Slide 12 of 21) Multiplication Law: Multiplication law can be used to calculate the probability of the intersection of two events. Based on the definition of conditional probability. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 34 Conditional Probability (Slide 13 of 21) Special case in which events A and B are independent. To compute the probability of the intersection of two independent events, simply multiply the probabilities of each event. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 35 Conditional Probability (Slide 14 of 21) Bayes’ Theorem: Often begin the analysis with initial or prior probability estimates for specific events of interest. Then, obtain additional information about events. Given new information, update the prior probability values by calculating revised probabilities, referred to as posterior probabilities. Bayes’ theorem provides a means for making these probability calculations. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 36 Conditional Probability (Slide 15 of 21) Example: A manufacturing firm receives shipments of parts from two different suppliers: 65% of the parts purchased from supplier 1. 35% of the parts purchased from supplier 2. Quality of purchased parts varies according to their source. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 37 Conditional Probability (Slide 16 of 21) Table 4.6: Historical Quality Levels for Two Suppliers Historical data suggest the quality ratings of the two suppliers: % Good Parts % Bad Parts Supplier 1 98 2 Supplier 2 95 5 Figure 4.7 shows a diagram that depicts the process of the firm receiving a part from one of the suppliers and then discovering that the part is good or bad as a two-step random experiment. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 38 Conditional Probability (Slide 17 of 21) Figure 4.7: Diagram for Two-Supplier Example Step 1 shows that the part comes from one of two suppliers and Step 2 shows whether the part is good or bad. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 39 Conditional Probability (Slide 18 of 21) The process of computing joint probabilities can be depicted in what is called a probability tree. From left to right through the tree: The probabilities for each branch at step 1 are prior probabilities. The probabilities for each branch at step 2 are conditional probabilities. To find the probability of each experimental outcome, multiply the probabilities on the branches leading to the outcome. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 40 Conditional Probability (Slide 19 of 21) Figure 4.8: Probability Tree for Two-Supplier Example © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 41 Conditional Probability (Slide 20 of 21) Suppose the parts from the two suppliers are used in the firm’s manufacturing process and a machine breaks while attempting the process using a bad part: Given the information that the part is bad, what is the probability that it came from supplier 1 and what is the probability that it came from supplier 2? With the information in the probability tree, Bayes’ theorem can be used to answer these questions. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 42 Conditional Probability (Slide 21 of 21) Bayes’ theorem is applicable when events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 43 Random Variables Discrete Random Variables Continuous Random Variables © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 44 Random Variables (Slide 1 of 6) In probability terms, a random variable is a numerical description of the outcome of a random experiment. Random variables are quantities whose values are not known with certainty. A random variable can be classified as being either: Discrete. Continuous. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 45 Random Variables (Slide 2 of 6) Discrete Random Variables: A random variable that can take on only specified discrete values is referred to as a discrete random variable. Table 4.7 provides examples of discrete random variables. Table 4.8 repeats the joint probability table for the Lancaster Savings and Loan data, but with the values labeled as random variables. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 46 Random Variables (Slide 3 of 6) Table 4.7: Examples of Discrete Random Variables Random Experiment Random Variable (x) Possible Values for the Random Variable Flip a coin Face of a coin showing 1 if heads; 0 if tails Roll a die Number of dots showing on top of die 1, 2, 3, 4, 5, 6 Contact five customers Number of customers who place an order 0, 1, 2, 3, 4, 5 Operate a health care clinic for one day Number of patients who arrive 0, 1, 2, 3, … Offer a customer the choice of two products Product chosen by customer 0 if none; 1 if choose product A; 2 if choose product B © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 47 Random Variables (Slide 4 of 6) Table 4.8: Joint Probability Table for Customer Mortgage Prepayments © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 48 Random Variables (Slide 5 of 6) Continuous Random Variables: A random variable that may assume any numerical value in an interval or collection of intervals is called a continuous random variable. Technically, relatively few random variables are truly continuous; examples are values related to time, weight, distance, and temperature. Many discrete random variables have a large number of potential outcomes and so can be effectively modeled as continuous random variables. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 49 Random Variables (Slide 6 of 6) Table 4.9: Examples of Continuous Random Variables © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 50 Discrete Probability Distributions Custom Discrete Probability Distribution Expected Value and Variance Discrete Uniform Probability Distribution Binomial Probability Distribution Poisson Probability Distribution © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 51 Discrete Probability Distributions (Slide 1 of 21) The probability distribution for a random variable describes the range and relative likelihood of possible values for a random variable. For a discrete random variable x, the probability distribution is defined by the probability mass function, denoted by The probability mass function provides the probability for each value of the random variable. We can present probability distributions graphically. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 52 Discrete Probability Distributions (Slide 2 of 21) Figure 4.9: Graphical Representation of the Probability Distribution for Whether a Customer Defaults on a Mortgage © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 53 Discrete Probability Distributions (Slide 3 of 21) Custom Discrete Probability Distribution: A probability that is generated from observations is called an empirical probability distribution. An empirical probability is considered a custom discrete probability distribution if it is discrete and the possible values of the random variable have different values: Useful for describing different possible scenarios that have different probabilities. Probabilities generated using either the subjective method or the relative frequency method. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 54 Discrete Probability Distributions (Slide 4 of 21) Table 4.10: Summary Table of Number of Payments Made per Year Example: The random variable describing the number of mortgage payments made per year by randomly chosen customers. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 55 Discrete Probability Distributions (Slide 5 of 21) Figure 4.10: Excel PivotTable for Number of Payments Made per Year © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 56 Discrete Probability Distributions (Slide 6 of 21) Expected Value and Variance: The expected value, or mean, of a random variable is a measure of the central location for the random variable. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 57 Discrete Probability Distributions (Slide 7 of 21) Table 4.11: Calculation of the Expected Value for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer If Lancaster Savings and Loan signs a new mortgage customer, the expected number of payments per year for this customer is 13.8. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 58 Discrete Probability Distributions (Slide 8 of 21) Figure 4.11: Using Excel SUMPRODUCT Function to Calculate the Expected Value for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 59 Discrete Probability Distributions (Slide 9 of 21) Figure 4.12: Excel Calculation of the Expected Value for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 60 Discrete Probability Distributions (Slide 10 of 21) Variance is a measure of variability in the values of a random variable: © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 61 Discrete Probability Distributions (Slide 11 of 21) Table 4.12: Calculation of the Variance for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 62 Discrete Probability Distributions (Slide 12 of 21) Figure 4.13: Excel Calculation of the Variance for Number of Payments Made per Year by a Lancaster Savings and Loan Mortgage Customer © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 63 Discrete Probability Distributions (Slide 13 of 21) Discrete Uniform Probability Distribution: When the possible values of the probability mass function are all equal, then the probability distribution is a discrete uniform probability distribution. Where n = the number of unique values that may be assumed by the random variable. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 64 Discrete Probability Distributions (Slide 14 of 21) Binomial Probability Distribution: A binomial probability distribution is a discrete probability distribution that can be used to describe many situations in which a fixed number (n) of repeated identical and independent trials has two, and only two, possible outcomes: Success. Failure. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 65 Discrete Probability Distributions (Slide 15 of 21) The probability mass function for a binomial random variable that calculates the probability of x successes in n independent events. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 66 Discrete Probability Distributions (Slide 16 of 21) Table 4.13: Probability Distribution for the Number of Customers Who Click on the Link in the Martin’s Targeted E-Mail Example: Martin’s, an online specialty clothing store, sends out targeted e-mails to its best customers notifying them about special discounts available only to the recipients. © 2021 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 67 Discrete Probability Distributions (Slide 17 of 21) Figure 4.14: Graphical Representation of the Probability Distribution for the Number of Customers Who Click on the Link in the Martin’s Targeted E-Mail © 2021 Cengage Learning. All …