$0.00 $0.50 $1.00 $1.50 $2.00 $2.50 $3.00 $3.50 $4.00 Price

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1 Orange Juice Sales and Prices In this module, you will be looking at sales and price data for orange juice in grocery stores. You have data from 83 stores on three brands (Tropicana, Minute Maid, and the store brand, Dominick s) over about two years. Note: selected answers are on the last page. Part 1: Price vs. Quantity Scatter Plots and Correlations 1) Tropicana at Store 5 Here is a scatter plot for the price and quantity pairs for Tropicana at Store 5 for the 116 weeks of data available. A (modified) version of this scatter plot shows on the tab called Tropicana Store 5. Units 100,000 80,000 Tropicana at Store 5 60,000 40,000 20,000 0 $0.00 $0.50 $1.00 $1.50 $2.00 $2.50 $3.00 $3.50 $4.00 Price Question: What is the correlation between price and quantity for Tropicana at Store 5? 2) Minute Maid at Store 5 a. Create a similar scatter plot that shows the price and quantity pairs for Minute Maid at Store 5. You may want to make a copy of the full data set (either by copying the OJ Data tab or the Tropicana Store 5 tab) and then filter it and graph it. b. What is the correlation between price and quantity for Minute Maid at Store 5? You should be able to answer these questions for any OJ brand and store pairing. 1

2 Part 2: Estimating the Price and Quantity Relationship for Tropicana at Store 5 Above we looked at the scatter plot of price and quantity for Tropicana in Store 5. In the chart below, you see a line that passes through the cloud of points. About half the points are above the line and about half of them are below it. Units 100,000 80,000 Tropicana at Store 5 60,000 40,000 20,000 0 $0.00 $0.50 $1.00 $1.50 $2.00 $2.50 $3.00 $3.50 $4.00 Price The line above intersects the horizontal axis at ($3.20, 0). In other words, on that line, for a price of $3.20, the quantity estimated by the line is 0. 1) Looking at the graph above and using the data in the spreadsheet, find the equation of this line. Hint: you can find the equation of a line if you know two points on the line. One of the points on this line is ($3.20, 0). What is another one? Look at the data to find the exact values. You should be able to determine the equation of a line through any two specific data points. 2

3 2) Look at the tab called Tropicana Store 5. On that tab, there is a graph with a scatter plot of price and quantity combined with a line graphed from an equation of quantity as a function of price. a. If you type in 110,000 for the intercept and 32,000 for the slope, what is the value for the line for a price of $2.00? You should be able to determine this estimated quantity for any intercept, slope, and price value. b. Adjust the intercept (b) and slope (m) values in the spreadsheet until you have what looks to you from your graph like a best fit. What are your intercept (b) and slope (m) values? b = m = c. Notice that the tab called Tropicana Store 5 also has a column for Estimated Quantity and one for Squared Difference. The Estimated Quantity is the y value of the line (y=mx+b) where the m and the b are in cells on the spreadsheet, and the x is the price value for that row. The Squared Difference is the squared difference between the (actual) Quantity Sold and the Estimated Quantity. Looking at the graph on the tab Tropicana Store 5, how would you describe what Squared Difference represents? d. There is also a cell on that tab that shows Total Squared Difference (i.e., the sum of the values in the Squared Difference column). The lower that total is, the better the fit of the line to the points. What is Total Squared Difference when intercept = 110,000 and slope = 32,000? e. What is the Total Squared Difference when intercept = 120,000 and slope = 35,000? 3

4 f. Find the values for intercept and slope that make the Total Squared Difference as low as possible. (Hint: try using the Solver tool in Excel. This is an add in and you may need to add it in.) What are the values of intercept (b) and slope (m) that minimize Total Squared Difference? b= m= 4

5 Part 3: Estimating the Price and Quantity Relationship for Minute Maid at Store 5 Now you are going to do a very similar exercise as above finding a good fit at first, and then a best fit, for a line through the price and quantity data for Minute Maid at Store 5. Look at the Price and Quantity scatter plot for Minute Maid for Store 5 that you created in Part 1. You should be able to do the following analysis for any OJ brand and store pairing. 1) Either digitally or on a printout, sketch a line that passes through the cloud of points. Your line should have about half the points above it and about half the points below, following approximately the overall trend of the cloud of points. Don t worry about getting exactly half above and below. 2) What are two points on your line? (, ) and (, ). 3) Find the equation for the line that passes through those two points, in y = mx + b form. b = m = The next set of questions will be much easier if you make a copy of the Tropicana Store 5 tab and change the filter in the Brand column to show Minute Maid instead of Tropicana. If you work with a copy, a lot of the calculations you need are already set up for you. It s a good idea to change the name of the new tab you created (to something like Minute Maid Store 5). 4) To check to make sure you got the tab copied and filtered correctly, answer the following questions about Minute Maid at Store 5: a. What was the price of Minute Maid in Store 5 in week 54? b. What was the quantity sold of Minute Maid in Store 5 in week 70? c. After you changed the filter on Brand, how many points remained on the graph on the tab? 5) Put the b value (intercept) and m value (slope) that you calculated in Question 3 into the appropriate cells on the spreadsheet for Minute Maid Store 5. The graph on the sheet should show the y=mx+b line. 5

6 6) If you haven t done so already, change the data for the graph (try to find Select Data and add a series) to show a scatter plot of the price and quantity pairs for Minute Maid at Store 5. 7) Change the Total Squared Difference cell to add up the Squared Difference values for Minute Maid at Store 5. Try using the dropdowns for Brand and Store # above Total Squared Difference. To see if you have that formula right, put Intercept (b) equal to and Slope (m) equal to What is the value of Total Squared Difference? 8) Find the values for intercept (b) and slope (m) that make the Total Squared Difference as low as possible. (Hint: try using the Solver tool in Excel.) What are the values of intercept (b) and slope (m) that minimize Total Squared Difference? b= m= 9) Optional/Extra: The steps above are the building blocks for running a simple regression, which is procedure for finding the best fit y as a linear function of x by minimizing the sum of squared deviations between the points and the line. There are (at least) two other ways to find that best fit line in Excel. See if you can get them to replicate your optimal m and b. a. Look for an option to add a Trendline to your scatter plot. There should also be a formatting option for the Trendline to display to equation of the line, if it doesn t display by default. b. Also super useful is the function for a linear regression, called LINEST() ( linear estimate ). In this case, you want to regress Quantity Sold on Price. (You will almost certainly have to look online for help on using the LINEST function.) 6

7 Part 4: Price and Profit You will calculate the store s profit (= (price cost) * quantity sold) for each product for each week. Assume that the cost of Tropicana is $0.95. (That would be the price that the store pays the company that makes Tropicana.) Assume that the cost of Minute Maid is $0.89. You should be able to do this analysis for any OJ Brand and Store pairing. 1) For Tropicana at Store 5, a. Create a scatter plot of price (on the horizontal axis) and profit (on the vertical axis). b. What was the price during the week with the highest profit? 2) For Minute Maid at Store 5, a. Create a scatter plot of price (on the horizontal axis) and profit (on the vertical axis). b. What was the price during the week with the highest profit? 3) For Tropicana at Store 5, a. Calculate estimated profit for price levels of $1.00, $1.50, $2.00,, $4.00. Estimated profit = (price cost) * estimated quantity. For estimated quantity, use quantity as predicted by the best fit line from the end of Part 2. (Those come from the m and b values that minimized Total Squared Difference.) What is the estimated profit for a price of $3.00? b. Write a function (by hand) for the weekly estimated profit f(x) (in $) when the price is x dollars. 7

8 c. Create a graph that has both the data in a scatter plot and the curve for estimated profit. d. Looking at the curve, what was the price associated with the highest estimated profit? 4) For Minute Maid at Store 5, a. Calculate estimated profit for price levels of $1.00, $1.50, $2.00,, $4.00. Estimated profit = (price cost) * estimated quantity. For estimated quantity, use quantity as predicted by the best fit line from the end of Part 3. (Those come from the m and b values that minimized Total Squared Difference.) What is the estimated profit for a price of $3.00? b. Write a function (by hand) for the weekly estimated profit f(x) (in $) when the price is x dollars. c. Create a graph that has both the data in a scatter plot and the curve for estimated profit. d. Looking at the curve, what was the price associated with the highest estimated profit? 8

9 Selected Answers Part 1 2b) For Minute Maid at Store 5, correlation is negative : Part 2 For another example, Dominick s at Store 8 is negative : ) y = 44,842 x + 143,496 2a) At $2.00, the estimated quantity is 46,000. 2d) Total Squared Difference = 21,558,639,616 for intercept = 110,000 and slope = 32,000 2e) Total Squared Difference = 25,768,683,516 for intercept = 120,000 and slope = 35,000 2f) The Total Squared Difference is minimized when the intercept is approximately 66,376 and the slope is approximately 17,951. Part 3 4a) The price of Minute Maid in Week 54 at Store 5 is $2.99 4b) The quantity of Minute Maid sold in Week 70 at Store 5 is 6,080. 7) Total Squared Difference = 29,746,743,040, for intercept = 60,000 and slope = 20,000 8) The Total Squared Difference is minimized when the intercept is approximately 74,736 and the slope is approximately 25,191. Part 4 1b) The price during the week with the highest profit is $2.49. (Profit is $141,631; Week is 143.) 2b) The price during the week with the highest profit is $1.99. (Profit is $109,894; Week is 128.) 3a) At $3.00 the estimated profit $25,675. 3b) Using the approximate values from Part 2 Question2f, the function is f(x) = (x 0.95)( 17,951x + 66,376) = 17951x x d) The high point of the curve is at about $ a) At $3.00 the estimated profit is approximately $ b) Using the approximate values from Part 3 Question 8, f(x) = (x 0.89)( x ) 4d) The high point of the curve is at about $