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Activity 2 - 5

Activity 2 - 5. Predicting Population. Objectives. Write an equation for a linear function give its slope and y-intercept Write linear functions in slope-intercept form, y = m x + b Determine the relative error in a measurement or prediction using a linear model

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Activity 2 - 5

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  1. Activity 2 - 5 Predicting Population

  2. Objectives • Write an equation for a linear function give its slope and y-intercept • Write linear functions in slope-intercept form, y = mx + b • Determine the relative error in a measurement or prediction using a linear model • Interpret the slope and y-intercept of linear functions in contextual situations • Use the slope-intercept form of linear equations to solve problems

  3. Vocabulary • Error – is the difference between the actual value and the predicted value; error = actual - predicted • Observed Value – also known as the actual value • Expected Value – also known as the predicted value • Relative Error – is the ratio of the error to the observed value

  4. P Activity According to the US Bureau of the Census, the population of the United States was approximately 132 million in 1940 and 151 in 1950. Assume that the rate of change of the population with respect to time is a constant value over the decade from 1940 to 1950. Write the data as two ordered pairs Plot the two data points and drawa line through them t 200 (1940, 132M) and (1950, 151M) 150 125 100 1950 1940

  5. Activity cont What is the average rate of change of the population from 1940 to 1950 (time t = 0 to t = 10)? What is the slope of the line? What is the practical meaning of the slope in the model? What is the P-intercept of this line? What is the practical meaning of the intercept? 151 – 132 19 ------------- = ----- = 1.9 million per year 10 – 0 10 m = 1.9 During that decade the population increased by an average of 1.9 million people each year With 1940 as t=0, 132 million is the P-intercept The population in 1940 was 132 million

  6. Activity cont 2 What is the slope-intercept form of the population model line? Assume that the average rate of change holds true through 1960. Use this equation to predict the 1960 US population. p = 1.9 (t) + 132 or p = 1.9 ( t – 1940) + 132 p = 1.9 (20) + 132 or p = 1.9 ( 1960 – 1940) + 132 p = 38 + 132 = 170 or p = 1.9(20) + 132 = 170

  7. Error and Relative Error • The error in a prediction is the difference between the observed value (actual value that was measured) and the predicted value from the model. The Relative error is the ratio of the error to the observed value. • It is normally expressed as a percentage Observed value – predicted value Relative error = -------------------------------------------------- Observed value error = ------------------------- Observed value

  8. Activity cont 3 The actual population for the US in 1960 was approximately 179 million. What is the relative error in the prediction we made? What do you think was the cause of the model’s error? Why would that be true (History question!)? 179 – 170 = 9 (error) 9 / 170 = 5.29% (relative error) Population growth was not a linear function Baby-boom generation after WWII

  9. y Population Model Lets look at more recent data. The US population was approximately 249 million in 1990 and 281 million in 2000. Determine the slope of this line Compare it with the slope of the1940 to 1950 model Which decade had the greatest population increase? x 281 – 249 32 ------------- = ----- = 3.2 million per year 10 – 0 10 The 90’s had 32 million compared to 19 million in the 40’s

  10. Population Model cont Determine the P-intercept in the new line Write the slope-intercept form of the equation of the line Predict the population in 2010 What assumption did we have to make in the prediction? If t = 0 in 1990, then P-intercept is 249 p = 3.2t + 249 p(20) = 3.2(20) + 249 = 313 million (Current Estimate: 310,354,711) That population growth is a linear function

  11. Population Model cont According to the linear model, P(t) = 3.2 t + 249, in what year will the population be 350 million? 350 = 3.2 t + 249 101 = 3.2 t 31.5625 = t So approximately July of 2021

  12. Summary and Homework • Summary • Error is the difference between actual and predicted values • Relative error is the ratio of the error to the observed value • Relative error is usually reported as a percentage • Homework • Pg 221-4; 1 - 6

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