IB Math Studies Internal Assessment. What is the Relationship between the Number of Legos in a Set and the Price of the Set? Exam Session: May 2017

Transcription

1 IB Math Studies Internal Assessment What is the Relationship between the Number of Legos in a Set and the Price of the Set? Exam Session: May 2017 Southwest Minneapolis High School Teacher: Mr Towne Date: 18 November 2016 William Towne

2 Introduction: While reading my son s Lego magazine, I saw that there was a contest to win a 3000 piece Lego set that wasn t sold in stores. I wondered how much it would cost if it was indeed sold. I know there must be some relationship between the number of Legos in a set and its cost, and my project will help me find it. I will also find out if there is a difference in the costs between different themes (Chima, Star Wars, Friends, etc.) My data is from Lego s website. On the website it lists the most recent sets by theme. I will write record for the latest 18 sets per theme: the theme the identification number of the set the name of the set the number of pieces in the set the cost of the set After collecting the data, I will first create a scatterplot of the number of pieces and the cost of the set. This will provide me with a description of the relationship between the two variables. Secondly, I will calculate the correlation between the number of pieces and the cost so that I can write about the reliability of the guess of the cost based on the number of pieces. Thirdly, I will find the line of best fit between pieces and cost so that I can use it to estimate the cost of the 3000 piece set. From the line of best fit I can also find out the cost of each brick and the cost of the packaging. Lastly, I will find separate lines of best fit for each theme to see if some themes cost more or less than other sets. Information There are many, many sets of Legos, but my data consists of 96 of the most recent sets. The data was taken from shop.lego.com. I chose six different themes of Lego sets: Star Wars, City, Friends, Ninjago, Classic, and Creator. I took the most recent 18 sets from each theme (they were ordered chronologically) since they were current. Some themes didn t have eighteen sets listed, so I chose what they did list. I chose the six themes based on what I perceived as being popular and they didn t include any sets that would have electronics since they would be priced more expensively and the 3000 piece set that I d like to know the cost of doesn t have any electronics. Here is a sample of the data, the rest of the data is in the appendix. Set ID Pieces Description Star Wars Death Star Star Wars Rebel Combat Frigate Star Wars Vader's TIE Advanced vs. A-Wing Starfighter City Volcano Heavy-Lift Helicopter City Volcano Exploration Base City Police Pursuit City Garbage Truck

3 Mathematical Processes From the scatter diagram below, I can see that the relationship between the number of pieces in a Lego set and the price of the set is a strong, positive, linear one. I can see that the relationship seems to be stronger if the number of pieces in the Lego set is less than After that, the prices seem to follow the same pattern but they are not nearly as strongly related. of Lego Sets Star Wars City Friends Ninjago Classic Creator The correlation, r, between the number of legos in a set and the price of the set is r r n xy x y n x x n y y r r r.907 This confirms that the relationship is strong (since the correlation is near one) and positive (since r > 0). The line of best fit, y = mx + b, are

4 2 2 2 n xy x y m n x x y mx b n Price = (Pieces) of Lego Sets y = x Star Wars City Friends Ninjago Classic Creator Interpretation of Results The equation tells me even more about the relationship. The slope of the line, is the price per brick of the Lego sets. That is, each brick costs, on average, about 8.41 cents. The y-intercept of the line is 12.1 is the cost of a zero brick set. That probably represents the costs that go into making the sets that aren t the physical bricks themselves. For instance, this could mean the development (including the directions), packing, and shipping costs. With this equation I can also estimate the cost of the 3000 piece set: Price = (3000) = $264 Since the prices of the sets all end in a.99 and generally go up by $5 increments, I would estimate a price of $259.99, or $264.99, or $

5 The set would appear on my line of best fit like shown above. Since the points for the larger sets can be far away from the line, I know that my estimate could be off. For instance, the 4000 piece set, using my line of best fit, would cost $350. But the actual set, the Star Wars Death Star, sells for $ So that one is off by $150 which is quite a large error. On the other hand, Lego Creator sets like Big Ben and Tower Bridge sell for $112 and $133 less than what the line would predict. Those are the two dots on the right side of the scatterplot above that are below the line of best fit. Finally, I notice that by looking at the themes separately, there are different types of patterns. So I did a scatterplot and the line of best fit for each theme, in Appendix A. Using these lines of best fit, I can find the cost per brick and the base cost for each theme. Theme per brick Base cost for the set Star Wars $ 3.37 City $ 4.54 Friends 8.54 $ 7.66 Ninjago 9.83 $ 3.19 Classic 6.13 $ 2.51 Creator 4.99 $ The cost per Lego does change for the different types of sets which I found interesting. Perhaps some types of sets have more elaborate, specific pieces that needs to be made especially for that type whereas other themes have mostly the basic pieces. I also noticed that the Creator set base cost was much larger than the other themes. I thought that there was something special about the way that they were made or boxed but couldn t figure out why after doing some research. So I added in more sets from 2015 which cost much less than the current lineup and found that they were actually much more in line with the other themes that I looked at. This data is shown in Appendix B. This probably occurred because in the original data, the cheapest set was quite expensive at $100. Theme per brick Base cost for the set Star Wars $ 3.37 City $ 4.54 Friends 8.54 $ 7.66 Ninjago 9.83 $ 3.19 Classic 6.13 $ 2.51 Creator* 6.80 $ 6.23 * with various 2014 and 2015 models Validity I began this project trying to find the price of a 3000 piece Lego set that I saw in a contest. I wanted to find other sets of Legos so that I could find out the price structure that is, how much do different sizes of Lego sets cost. To do this, I needed a line of best fit. But in order to find the line of best fit, I first had to be sure that the data was in fact linear, which I found it was using a scatterplot.

6 But there are some possible warnings. First, the set that was in a contest didn t conform to one of the themes of Legos that is sold in stores. Because the lines of best fit for the different themes are slightly different, my estimated price may be incorrect. I believe that these are slightly different because of licensing fees or more elaborate bricks for certain sets. Without knowing what theme or what types of bricks were in the 3000 piece set, my guess may be inaccurate. In some of the larger models, the actual cost is quite far from the estimated cost, even when using interpolation. Perhaps if I knew what percentage of bricks in each set was standard rather than custom made, I could better fit my data, but this would introduce another variable. A second problem could be lack of data in my set. I only used six themes, but there are many more. I chose the themes I did because there are many sets from each, but I could have used more data in my scatterplot.

7 Appendix A Scatterplot of Number of Legos versus for different themes of Legos. Star Wars Linear (Star Wars) City Linear (City) y = x R² = y = x R² = Friends Linear (Friends) Ninjago Linear (Ninjago) y = x R² = y = x R² = Classic Linear (Classic) Creator Linear (Creator) y = x R² = y = x R² =

8 Appendix B - Scatterplot of Number of Legos versus for the Creator theme before and after adding, cheaper and smaller, 2015 sets Sets (n = 10) 2015 and 2016 Sets (n = 34) Creator Linear (Creator) Creator Linear (Creator) y = x R² = y = 0.068x R² = Appendix C Sums for calculating Pearson s Correlation Coefficient and the Regression Line ΣPieces Σ ΣPieces 2 Σ 2 Σ (Pieces)()

9 Appendix D Raw data Set ID Pieces Description Predicted Error Star Wars Death Star Star Wars Rebel Combat Frigate Star Wars Vader's TIE Advanced vs. A-Wing Starfighter Star Wars Krennic's Imperial Shuttle Star Wars Rebel U-Wing Fighter Star Wars TIE Striker Star Wars Encounter on Jakku Star Wars AT-ST Walker Star Wars Imperial Assault Hovertank Star Wars Sergeant Jyn Erso Star Wars K-2SO Star Wars Imperial Death Trooper Star Wars Sandcrawler Star Wars Ewok Village Star Wars Assault on Hoth Star Wars Slave I Star Wars TIE Fighter Star Wars Millennium Falcon City Volcano Heavy-Lift Helicopter City Volcano Exploration Base City Police Pursuit City Garbage Truck City Ambulance Plane City x4 Off Roader City Volcano Starter Set City Fire Starter Set City Rally Car City Prison Island Starter Set City Race Boat City Tire Escape City Fire ATV City Heavy-Haul Train City Cargo Train City City Square City High-Speed Passenger Train City Deep Sea Exploration Vessel Friends Amusement Park Roller Coaster Friends Heartlake Performance School Friends Amusement Park Bumper Cars Friends Amusement Park Hot Dog Van

10 Set ID Pieces Description Predicted Error Friends Heartlake Party Shop Friends Heartlake Puppy Daycare Friends LEGO Friends Advent Calendar Friends Amusement Park Space Ride Friends Amusement Park Arcade Friends Foal's Washing Station Friends Party Styling Friends Heartlake Grand Hotel Friends Heatlake Airport Friends Adventure Camp Tree House Friends Emma's House Friends Heartlake Riding Club Friends Pop Star Tour Bus Friends Livi's Pop Star House Ninjago Temple of Airjitzu Ninjago Samurai X Cave Chaos Ninjago Final Flight of Destiny's Bounty Ninjago Ultra Stealth Raider Ninjago City of Stiix Ninjago Misfortune's Keep Ninjago The Lighthouse Siege Ninjago Attack of the Morro Dragon Ninjago Airjitzu Battle Grounds Ninjago Titan Mech Battle Ninjago The Green NRG Dragon Ninjago Tiger Widow Island Ninjago Ronin R.E.X Ninjago Salvage M.E.C Ninjago Rock Roader Ninjago Jay Walker One Ninjago Master Wu Dragon Ninjago Jay's Elemental Dragon Classic LEGO Creative Building Basket Classic LEGO Large Creative Brick Box Classic LEGO Medium Creative Brick Box Classic LEGO Creative Building Set Classic LEGO Creative Supplement Classic LEGO Creative Supplement Bright Classic LEGO Creative Bricks Creator Fast Car Creator Ocean Explorer Creator Park Animals

11 Set ID Pieces Description Predicted Error Creator Super Soarer Creator Chopper Transporter Creator Construction Vehicles Creator Desert Racers Creator Sydney Opera House Creator Big Ben Creator Tower Bridge Creator Ferris Wheel Creator Detective's Office Creator Parisian Restaurant Creator Fairground Mixes Creator Palace Cinema Creator Pet Shop Creator Volkswagen Beetle