1. Copyright © 2014 Pearson Education. All rights reserved. 2.3-1 2.3 Uses of Percentages in Statistics LEARNING GOAL Understand how percentages are used to report statistical results and recognize ways in which they are sometimes misused.

2. Copyright © 2014 Pearson Education. All rights reserved. 2.3-2 Slide 2.3- 2 Conversions Between Fractions and Percentages To convert a percentage to a common fraction: Replace the % symbol with division by 100; simplify the fraction if necessary. To convert a percentage to a decimal: Drop the % symbol and move the decimal point two places to the left (that is, divide by 100).

3. Copyright © 2014 Pearson Education. All rights reserved. 2.3-3 Slide 2.3- 3 Conversions Between Fractions and Percentages To convert a decimal to a percentage: Move the decimal point two places to the right (that is, multiply by 100) and add the % symbol. To convert a common fraction to a percentage: First convert the common fraction to a decimal; then convert the decimal to a percentage.

4. Copyright © 2014 Pearson Education. All rights reserved. 2.3-4 Slide 2.3- 4 EXAMPLE 1 Newspaper Survey A newspaper reports that 44% of 1,069 people surveyed said that the President is doing a good job. How many people said that the President is doing a good job?

5. Copyright © 2014 Pearson Education. All rights reserved. 2.3-5 Slide 2.3- 5 Using Percentages to Describe Change Absolute and Relative Change The absolute change describes the actual increase or decrease from a reference value to a new value: absolute change = new value – reference value The relative change describes the size of the absolute change in comparison to the reference value and can be expressed as a percentage: relative change = × 100% new value – reference value reference value

6. Copyright © 2014 Pearson Education. All rights reserved. 2.3-6 Slide 2.3- 6 EXAMPLE 2 World Population Growth World population in 1950 was 2.6 billion. By the beginning of 2000, it had reached 6.0 billion. Describe the absolute and relative change in world population from 1950 to 2000.

7. Copyright © 2014 Pearson Education. All rights reserved. 2.3-7 Slide 2.3- 7 Using Percentages for Comparisons Percentages are also commonly used to compare two numbers. In this case, the two numbers are the reference value and the compared value. The reference value is the number that we are using as the basis for a comparison. The compared value is the other number, which we compare to the reference value.

8. Copyright © 2014 Pearson Education. All rights reserved. 2.3-8 Slide 2.3- 8 Absolute and Relative Difference The absolute difference is the difference between the compared value and the reference value: absolute difference = compared value - reference value The relative difference describes the size of the absolute difference in comparison to the reference value and can be expressed as a percentage: relative difference = × 100% compared value - reference value reference value

9. Copyright © 2014 Pearson Education. All rights reserved. 2.3-9 Slide 2.3- 9 EXAMPLE 3 Russian and American Life Expectancy Life expectancy for American men is about 75 years, while life expectancy for Russian men is about 59 years. Compare the life expectancy of American men to that of Russian men in absolute and relative terms.

10. Copyright © 2014 Pearson Education. All rights reserved. 2.3-10 Slide 2.3- 10 “Of” versus “More Than” (or “Less Than”) If the new or compared value is P% more than the reference value, then it is (100 + P)% of the reference value. If the new or compared value is P% less than the reference value, then it is (100 - P)% of the reference value.

11. Copyright © 2014 Pearson Education. All rights reserved. 2.3-11 Slide 2.3- 11 EXAMPLE 4 World Population In Example 2, we found that world population in 2000 was about 131% more than world population in 1950. Express this change with an “of ” statement.

12. Copyright © 2014 Pearson Education. All rights reserved. 2.3-12 Slide 2.3- 12 Percentages of Percentages Percentage Points versus % When you see a change or difference expressed in percentage points, you can assume it is an absolute change or difference. If it is expressed as a percentage, it probably is a relative change or difference.

13. Copyright © 2014 Pearson Education. All rights reserved. 2.3-13 Slide 2.3- 13 Based on interviews with a sample of students at your school, you conclude that the percentage of all students who are vegetarians is probably between 20% and 30%. Should you report your result as “25% with a margin of error of 5%” or as “25% with a margin of error of 5 percentage points”? Explain. EXAMPLE 6 Margin of Error