Probability and Statistics

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District of Columbia Public Schools Page 1 of 13

Probability and Statistics

Week 2

DCPS Distance Learning Plan

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District of Columbia Public Schools

Probability and Statistics Learning Plan

5 minutes

Day 1 Opening

Directions: Solve the Opening in the space provided below. Please be sure to show all work!

Have you ever noticed how sometimes batteries seem to last a long time, and other times the batteries seem to last only a short time?

The histogram below shows the distribution of battery life (hours) for a sample of 40 batteries of the same brand.

When studying a distribution, it is important to think about the shape, center, and spread of the data.

1. Would you describe the distribution of battery life as approximately symmetric or as skewed? Explain your answer.

2. Is the mean of the battery life distribution closer to 95, 105, or 115 hours? Explain your answer.

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Act One: #PopularityContest

1 When an Instagram user posts a picture, they often care about the number of interactions – likes and comments – it receives. The graphs below are based on more than 10,000 randomly selected Instagram posts. Which factor seems most strongly related with how many interactions a post gets, and how would you describe the relationship?

Interactions vs. User Tags Interactions vs. Hashtags

y = 20.9x + 65.7 (R² = 0.0087) y = 0.57x + 87.8 (R² = 0.00005) Interactions vs. Followers Interactions vs. Minute of Day

y = 0.03x + 26.9 (R² = 0.6833) y = 0.03x + 74.7 (R² = 0.00027)

2 In addition to caring about how many interactions a post gets, users also care about how many followers they get.

If a user wanted to gain more followers on Instagram, what would you recommend they do and why?

Followers vs. ... ... Following ... Total Posts ... Days as User ... Avg. Posts per Day y = 1.14x + 1148.8

(R² = 0.0072)

y = 0.91x + 1443 (R² = 0.0097)

y = 1.49x + 427.4x (R² = 0.0047)

y = 151.4x + 2109 (R² = 0.00043) 0

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45 minutes

Day 1- Task

District of Columbia Public Schools Source: Mathalicious

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Act Two: #NoFilter?

3 Union Metrics helps companies like Nike and Airbnb use social media more effectively. The graph below shows the relationship between the percent of interactions a post receives and the time since it was posted. If you were advising a company how often it should post to Instagram, what would you recommend, and what might be some consequences of posting this often? What if you were advising a friend?

% of Total Interactions

Time Since Post

4 What might be some consequences of people feeling like they have to maintain a personal brand, and what do you expect would happen if more users took the “Instagram Challenge” to show what their lives were actually like?

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Correlations with Interactions (n = 10,534 posts)

Interactions vs. User Tags Interactions vs. Hashtags

y = 20.9x + 65.7 (R² = 0.0087) y = 0.57x + 87.8 (R² = 0.00005)

Interactions vs. Followers Interactions vs. Minute of Day

y = 0.03x + 26.9 (R² = 0.6833) y = 0.03x + 74.7 (R² = 0.00027) 0

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Source: Mathalicious

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Correlations with Followers (n = 10,534 posts)

Followers vs. Following Followers vs. Total Posts

y = 1.14x + 1148.8 (R² = 0.0072) y = 0.91x + 1443 (R² = 0.0097)

Followers vs. Days as User Followers vs. Avg. Posts per Day

y = 1.49x + 427.4 (R² = 0.0047) y = 151.4x + 2109 (R² = 0.00043) 0

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y = 21.195x + 2231.2 R² = 4E-05

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5 minutes

Day 1- Reflection

What is one thing you learned or found interesting today?

What is one question that you still have after today?

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5 minutes

Day 2 Opening

The histogram below shows the distribution of the greatest drop (in feet) for 55 major roller coasters in the United States.

1. Is the mean of the maximum drop distribution closer to 90, 135, or 240 feet? Explain your answer.

2. Is the standard deviation of the maximum drop distribution closer to 40, 70, or 100 hours? Explain your answer.

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Act One: Moneyball

1 All sports teams have the same goal: to win. However, not all teams have the same amount of money, and some fans think wealthier teams have an unfair advantage. The charts below show how much NFL, MLB, and NBA teams paid their players in the most recent seasons and how many regular-season games they won. Choose a sport and explain what the graph tells you. Does it appear that teams who spend more money win more games?

Wins (y) vs. Team Payroll, $millions (x)

2018-19 2018 2018-19

y = -0.035x + 15.3 | r² = 0.163 y = 0.088x + 67.2 | r² = 0.088 y = -0.104 + 56.1 | r² = 0.059

2 Even on the same team, some players make more than others do. If a team wanted to win more games, which do you expect would be more effective: paying the team more or paying a single superstar more? Explain.

Wins (y) vs. Salary of Highest-Paid Player, $millions (x)

y = -0.016x + 8.3 | r² = 0.004 y = 0.206x + 76.5 | r² = 0.009 y = 0.864x + 19.73 | r² = 0.317 0

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45 minutes

Day 2-Task

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Act Two: Buy a Win?

3 In pro sports, each regular-season game can prove the difference between going to the playoffs and going home.

Choose a sport, and imagine you’re an agent representing a superstar in a salary negotiation. On average, how much would you say the team would need to spend for each additional win?

Wins (y) vs. Salary of Highest-Paid Player, $millions (x)

y = -0.016x + 8.3 | r² = 0.004 y = 0.206x + 76.5 | r² = 0.009 y = 0.864x + 19.73 | r² = 0.317

4 The graphs below show the number of games NFL, MLB, and NBA teams won and the total revenues they earned from ticket sales, TV rights, etc. If you were the owner of the team negotiating with the agent, would you be willing to pay the amount above for an additional win? If so, why? If not, why do you think teams spend so much on payrolls?

Revenue, $millions (y) vs. Wins (x)

y = 5.11x + 370.6 | r² = 0.026 y = 2.91x + 79.5 | r² = 0.202 y = -0.75x + 297.5 | r² = 0.024

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5 minutes

Day 2- Reflection

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5 minutes

Day 3 Opening

Consider the following histograms: Histogram 1, Histogram 2, Histogram 3, and Histogram 4. Descriptions of four distributions are also given. Match the description of a distribution with the appropriate histogram.

Histogram Distribution

1 2 3 4 Description of distributions:

Distribution Shape Mean Standard Deviation

𝐴𝐴 Skewed to the right 100 10

𝐵𝐵 Approximately symmetric, mound shaped 100 10

𝐶𝐶 Approximately symmetric, mound shaped 100 40

𝐷𝐷 Skewed to the right 100 40

Histograms:

Histogram 3 Histogram 4

Histogram 1 Histogram 2

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Country GDP (trillions of USD) Latitude Medals (2006) Medals (2010)

Canada 1.574052 45.42 24 26

Germany 3.309669 52.52 29 30

United States 14.5824 38.90 25 37

Norway 0.414462 59.95 19 23

South Korea 1.014483 37.57 11 14

Switzerland 0.523772 46.95 14 9

China 5.878629 39.91 11 11

Sweden 0.458004 59.33 14 11

Austria 0.376162 48.20 23 16

Netherlands 0.783413 52.37 9 8

Russia 1.479819 55.75 22 15

France 2.560002 48.86 9 11

Australia 0.924843 35.31 2 3

Czech Republic 0.192152 50.08 4 6

Poland 0.468585 52.23 2 6

Italy 2.051412 41.90 11 5

Belarus 0.054713 53.90 1 3

Slovakia 0.089034 48.14 1 3

Great Britain 2.246079 51.51 1 1

Japan 5.497813 35.69 1 5

Croatia 0.060852 45.82 3 3

Slovenia 0.047763 46.06 0 3

Latvia 0.02401 56.95 1 2

Finland 0.238801 60.17 9 5

Estonia 0.018674 59.44 3 1

Kazakhstan 0.142987 51.17 0 1

45 minutes

Day 3-Task

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Your eyes are not deceiving you.

This page is supposed to be blank.

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Act One: But I Regress…

1 The scatterplot below shows a country’s Gross Domestic Product (a measure of its wealth) and the number of medals it won in the 2010 Winter Olympics. What does the data tell you about the relationship between GDP and performance in the Winter Games?

2010 Medals

GDP (trillions of dollars)

2 Japan’s GDP in 2010 was approximately $5.5 trillion. Based on this, how many medals would you predict that they won in the 2010 Winter Olympics? How confident are you in your prediction and why?

3 There are other factors besides a country’s wealth that may predict success at the Olympics. For each country that has won a medal at the Winter games, the table of data lists two other possible factors: the latitude of the country’s capital, and the number of medals the country won in 2006.

Use this data to come up with a line of best fit and correlation coefficient for each of these variables. Based on this, which variable – GDP, latitude, or medals earned in 2006 – do you think does the best job of predicting performance in the Olympics? Explain.

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y = 1.89x + 6.66 r ≈ 0.60

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Act Two: Model Behavior

4 For each linear model you’ve looked at, find an example of a country whose performance at the Olympics was close to the model’s predictions. Then, find and an example of a country whose performance wasn’t so close.

What do these examples suggest to you about using a single variable to predict Olympic performance?

5 Two brothers named Dan and Tim Graettinger developed a model to predict winter Olympic medals by combining several different factors (including some of the ones you looked at earlier). Below is a graph showing their predictions for the 2014 games, along with the actual results. How do their errors compare to those you saw previously? What do you think this means?

2014 Actual

2014 Predicted 0

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5 minutes

Day 3- Reflection

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5 minutes

Day 4 Opening

The histogram below shows the distribution of gasoline tax per gallon for the 50 states and the District of Columbia in 2010. Describe the shape, center, and spread of this distribution.

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Act One: Line It Up

1 In 2012, the New Economics Foundation released the Happy Planet Index report, which measured well-‐being in more than 150 countries. The lines of best fit describe the relationship between well-‐being and various measures. Pick one measure, e.g. life expectancy or governance. As the measure increases, how does well-‐being change? (The governance score accounts for factors like political stability, accountability, and the absence of corruption.)

Well-‐Being vs. Life Expectancy Well-‐Being vs. Median per Capita Income

y = 0.0895x – 0.9224 y = 0.0002x + 4.6787

Well-‐Being vs. Avg. Temperature in Capital Well-‐Being vs. Governance

y = -‐0.0314x + 7.3834 y = 0.1375x + 4.0809

2 According to the report, Finland performed better than did the United States in 2012 on well-‐being, life expectancy, median income, and governance. If the U.S. government were considering a policy that could match Finland’s performance in one of these areas, which do you think would be the most beneficial and why?

Well-‐Being Life Expectancy Median Income Governance

United States 7.2 78.5 years $15,480 17.4

Finland 7.4 80.0 years $15,725 21.2

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45 minutes

Day 4-Task

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Act Two: Scatterbrained

3 The graphs below show the actual data for the 151 countries covered in the Happy Planet Index report, including the correlation coefficients, r, for each line of best fit. Use this to answer the following questions.

Well-‐Being vs. Life Expectancy Well-‐Being vs. Median Income y = 0.0895x – 0.9224 r = 0.72 y = 0.0002x + 4.6787 r = 0.73

Well-‐Being vs. Avg. Temperature Well-‐Being vs. Governance y = -‐0.0314 + 7.3834 r = -‐0.37 y = 0.1375x + 4.0809 r = 0.67

a. Which measures are the most and least strongly

correlated with well-‐being, and why might this be? b. Many policymakers believe the most important goal is a strong economy. Do you agree?

4 Many people assume that if they make more money, they’ll be happier. The graphs below show the fraction of people who experience happiness and stress for different incomes, according to economist Daniel Kahneman.

Do you think more money translates to more happiness? If so, why? If not, what are some things that might?

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Probability and Statistics