1. Lesson 26 Making Inferences

2. We often need to use existing data to make an educated guess about the future. You might use your knowledge of how often a teacher gives a quiz in science to infer when the next quiz will be. Or you might use your favorite team’s record to infer whether the team will win the next game or the next five games. We often need to use existing data to make an educated guess about the future. You might use your knowledge of how often a teacher gives a quiz in science to infer when the next quiz will be. Or you might use your favorite team’s record to infer whether the team will win the next game or the next five games.

3. Example 1 John studied the data in this table which shows the changing population of his hometown. From these data, he estimated the population of his hometown in the year 2010. What is a good estimate of what the population will be in the year 2010? John studied the data in this table which shows the changing population of his hometown. From these data, he estimated the population of his hometown in the year 2010. What is a good estimate of what the population will be in the year 2010?

4. Strategy: Look for a pattern in the population data. Strategy: Look for a pattern in the population data. Step 1: Round the population to the nearest 500. Step 1: Round the population to the nearest 500.

5. Step 2: Find the differences of the population data estimates (Step 1) for each 10-year period. Step 2: Find the differences of the population data estimates (Step 1) for each 10-year period.

6. Step 3: Look for a pattern. The differences of Step 2 reveal a pattern.. Three of the five differences are 2000. The two exceptions are 500 from 2000. You can project based on a difference of 2000. Step 3: Look for a pattern. The differences of Step 2 reveal a pattern.. Three of the five differences are 2000. The two exceptions are 500 from 2000. You can project based on a difference of 2000.

7. Step 4:Estimate what the population will be in the year 2010. Using 2000 as an estimate of the differences between the population figures, add 2000 to the estimated population 14,500. Step 4:Estimate what the population will be in the year 2010. Using 2000 as an estimate of the differences between the population figures, add 2000 to the estimated population 14,500.

8. Solution The estimated population of John’s hometown in the year 2010 is 16,500. The estimated population of John’s hometown in the year 2010 is 16,500.

9. Example 2 Samantha collected data about the price of a popular chocolate bar at the local candy shop. She drew a graph of the prices in five-year intervals over a 20- year period. If the trend continues, what will be the price in 2005? Samantha collected data about the price of a popular chocolate bar at the local candy shop. She drew a graph of the prices in five-year intervals over a 20- year period. If the trend continues, what will be the price in 2005?

10. Strategy: Use a trend line to predict the future price. Strategy: Use a trend line to predict the future price. Step 1: Draw a line through he existing points of the graph. This line is called a trend line. It does not have to pass through all the points since it approximates the overall direction of the set of points. Step 1: Draw a line through he existing points of the graph. This line is called a trend line. It does not have to pass through all the points since it approximates the overall direction of the set of points.

11. Step 2: Extend the horizontal axis so that the year 2005 is represented. Draw the vertical line representing 2005. Step 2: Extend the horizontal axis so that the year 2005 is represented. Draw the vertical line representing 2005.

12. Step 4: Determine the price in the year 2005. The price is the point of intersection of the trend line and the vertical line for the year 2005. It is around 73 cents. Step 4: Determine the price in the year 2005. The price is the point of intersection of the trend line and the vertical line for the year 2005. It is around 73 cents.

13. Solution The price will be approximately 73 cents. The price will be approximately 73 cents.