According to the 2010 Census, the adult population aged 18 years or greater of the United States in 2010 was 234,564,071. In 2010, a survey was conducted among a randomly chosen sample of adults aged 18 years or greater in the United States about their preference to live in a warm climate or a cool climate. The table below summarizes the survey results.
| Climate Preferences | Warm | Cool | No preference | Total |
|---|---|---|---|---|
| 18–35 years old | 295 | 168 | 45 | 508 |
| 36–50 years old | 246 | 123 | 41 | 410 |
| 51–65 years old | 238 | 117 | 48 | 403 |
| Greater than 65 years old | 137 | 78 | 64 | 279 |
| Total | 916 | 486 | 198 | 1,600 |
Which of the following is closest to the difference between the percentage of adults aged 18–50 years who responded "warm" and the percentage of adults aged 51 years or greater who responded "warm"?
A conditional percentage gives the share within a named group, not within the entire survey. If an age range combines table rows, add its warm counts and row totals before dividing. Evaluate the two fractions in Desmos, then subtract the percentages. Dividing by the warm-column total instead flips the question: it measures the ages of warm respondents.
Hints
- Hint 1
An age range that spans two table rows needs both rows. For adults ages 18 to 50, add the warm counts for the part, then add the row totals for the whole. What fraction results?
- Hint 2
Apply the same part-over-whole rule to adults ages 51 and over. The whole is everyone surveyed in those two older rows, not everyone in the warm column. Which counts belong below the fraction bar?
- Hint 3
Convert each fraction to a percentage by multiplying by . Then subtract the two percentages to get a percentage-point difference, not a percent change based on one group.
Step-by-step
Build each age group's percentage
Step 1Find the warm share for ages 18 to 50
The 18-to-50 group combines the 18-to-35 and 36-to-50 rows. Add their warm responses above their row totals: . For each group's percentage, divide its warm count by its own total. The Census population isn't a count of surveyed adults.
- Step 2
Find the warm share for ages 51 and over
The 51-and-over group combines the 51-to-65 and greater-than-65 rows: . The denominator is this group's total. Using , the warm-column total, would instead tell you how warm respondents are split by age.
- Step 3
Calculate the younger group's percentage
Call the younger group's warm percentage . Type ; Desmos shows . Multiplying by changes the part-over-whole fraction into a percentage.
- Step 4
Calculate the older group's percentage
Call the older group's warm percentage . Add in Desmos; it shows . This uses all surveyed adults in the older group as the whole.
- Step 5
Compare the two percentages
Type ; Desmos prints about . That's a difference of about percentage points between the age groups' warm-response percentages. The closest listed difference is . Choice C.