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General interest: predicting the world record for running the mile. Note that \(x\) represents the actual year in following table. $$\begin{array}{|lc|}\hline & \text { World Record, } y \\\\\text { Year, } x & \text { (in minutes:seconds) } \\\1875 \text { (Walter Slade) } & 4: 24.5 \\\1894 \text { (Fred Bacon) } & 4: 18.2 \\\1923 \text { (Paavo Nurmi) } & 4: 10.4 \\\1937 \text { (Sidney Wooderson) } & 4: 06.4 \\\1942 \text { (Gunder Hägg) } & 4: 06.2 \\\1945 \text { (Gunder Hägg) } & 4: 01.4 \\\1954 \text { (Roger Bannister) } & 3: 59.6 \\\1964 \text { (Peter Snell) } & 3: 54.1 \\\1967 \text { (Jim Ryun) } & 3: 51.1 \\\1975 \text { (John Walker) } & 3: 49.4 \\\1979 \text { (Sebastian Coe) } & 3: 49.0 \\\1980 \text { (Steve Ovett) } & 3: 48.40 \\\1985 \text { (Steve Cram) } & 3: 46.31 \\\1993 \text { (Noureddine Morceli) } & 3: 44.39 \\\\\hline\end{array}$$ a) Find the regression line, \(y=m x+b,\) that fits the data in the table. (Hint: Convert each time to decimal notation; for instance, \(\left.4: 24.5=4 \frac{24.5}{60}=4.4083 .\right)\) b) Use the regression line to predict the world record in the mile in 2010 and in 2015 c) In July \(1999,\) Hicham El Guerrouj set the current (as of December 2006 ) world record of 3: 43.13 for the mile. (Source: USA Track \(\&\) Field and infoplease.com.) How does this compare with what is predicted by the regression line?

Step by step solution

01

Convert Times to Decimal Notation

Convert each world record time to decimal form. For example, 4:24.5 minutes converts to 4.4083 by computing \[ 4 + \frac{24.5}{60} \].

02

Prepare Data for Regression

List the years (x) and converted times (y) in two separate lists for further calculations.

03

Calculate Averages

Find the average of the years \( \bar{x} \) and the average of the times \( \bar{y} \).

04

Calculate the Slope (m)

Use the formula \[ m = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sum{(x - \bar{x})^2}} \] to calculate the slope of the regression line.

05

Calculate the Intercept (b)

Use the formula \( b = \bar{y} - m\bar{x} \) to find the y-intercept of the regression line.

06

Write the Regression Line Equation

Combine the slope (m) and intercept (b) into the equation \( y = mx + b \).

07

Predict Future Records

Substitute the years 2010 and 2015 into the regression equation to predict the world records for those years.

08

Compare with Actual 1999 Record

Calculate the predicted record for 1999 using the regression line and compare it to the actual record set by Hicham El Guerrouj in July 1999 of 3:43.13 minutes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression

Linear regression is a fundamental tool in statistics and machine learning.
It helps us identify relationships between variables and use them to make predictions.
In this exercise, we're using linear regression to predict world records in running the mile.
By finding a linear equation that best fits the data points, we can estimate future values.
The regression line is represented as the equation: \[ y = mx + b \] where \( y \) is the predicted value, \( x \) is the independent variable (years in our case), \( m \) is the slope, and \( b \) is the y-intercept.
Calculating this line involves several steps, as detailed in the solution provided.

Data Conversion

In many cases, your data won't initially be in a convenient format for analysis.
For this exercise, our times are given in minutes and seconds.
To perform linear regression, we need to convert these times into decimal format.
For instance, 4:24.5 minutes converts to 4.4083 minutes by computing: \[ 4 + \frac{24.5}{60} \] This conversion makes it easier to handle the data mathematically.
Accurate data conversion is crucial because it ensures that the subsequent calculations for slope and intercept are correct.

Predictive Modeling

Predictive modeling uses statistical techniques to predict future outcomes based on historical data.
Linear regression is one of the simplest forms of predictive modeling.
In this exercise, once we have our regression line, we can use it to predict world records for future years.
For example, by substituting the years 2010 and 2015 into our regression equation, we can estimate what the mile world record might be during those years.
Predictive models are widely used in various fields, such as finance, healthcare, and sports analytics.

Slope Calculation

The slope (m) of a regression line indicates the rate of change between the independent and dependent variables.
To calculate the slope, follow these steps:

• Find the average of the years (\( \bar{x} \)) and the average of the times (\( \bar{y} \)).
• Use the formula: \[ m = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{\sum{(x - \bar{x})^2}} \]

The slope tells us how much the world record time decreases per year.
A negative slope, as in this case, indicates that the times are generally getting faster over the years.

Intercept Calculation

The intercept (b) of a regression line is the point where the line crosses the y-axis.
It represents the value of the dependent variable when the independent variable is zero.
To calculate the intercept, use the formula: \[ b = \bar{y} - m\bar{x} \]
This value helps us complete the regression equation.
In our case, it would estimate the world record time at year zero, which may not be meaningful by itself but is necessary to form the complete linear equation.
With both slope and intercept, we form the regression line equation \( y = mx + b \) and use it for predictions.