3. A national health organization warns that 30% of the middle school students nationwide have been drunk. A local health agency surveys 110 of the 1212 middle school students in i... 3. A national health organization warns that 30% of the middle school students nationwide have been drunk. A local health agency surveys 110 of the 1212 middle school students in its city. Only 21 of them report having been drunk. a) What proportion of the sample reported having been drunk? b) Does this mean that this city’s youth are not drinking as much as the national data would indicate?
Understand the Problem
The question is asking to calculate the proportion of a sample of middle school students who reported having been drunk and to analyze whether this city's youth are drinking less compared to national statistics. This involves basic statistical analysis and interpretation of data.
Answer
a) The proportion is approximately 0.1909 or 19.09%. b) Yes, the city’s youth are drinking less than the national average.
Answer for screen readers
a) The proportion of the sample that reported having been drunk is approximately 0.1909 or 19.09%.
b) Yes, this indicates that this city’s youth are not drinking as much as the national data indicates.
Steps to Solve
- Calculate the Proportion of the Sample To find the proportion of the sample that reported having been drunk, you divide the number of students who reported being drunk by the total number of students surveyed.
The calculation is: $$ \text{Proportion} = \frac{\text{Number of students drunk}}{\text{Total students surveyed}} = \frac{21}{110} $$
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Perform the Proportion Calculation Calculating the proportion gives: $$ \text{Proportion} = \frac{21}{110} \approx 0.1909 $$ This means about 19.09% of the surveyed students reported having been drunk.
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Compare with National Data Next, compare the result with the national statistic. The national organization indicates that 30% have been drunk, which is: $$ \text{National Proportion} = 0.30 $$
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Analysis of Results Now we assess if the city's youth are drinking less. Since 19.09% (0.1909) is less than 30% (0.30), it suggests that the youth in this city report lower levels of drinking compared to the national average.
a) The proportion of the sample that reported having been drunk is approximately 0.1909 or 19.09%.
b) Yes, this indicates that this city’s youth are not drinking as much as the national data indicates.
More Information
The proportion indicates that in this specific survey, fewer middle school students reported having been drunk compared to the national average. This could suggest positive trends in youth drinking behaviors in this city.
Tips
- Misinterpreting Proportions: Confusing the percentage with raw numbers; always convert proportion to a percentage for clearer understanding.
- Neglecting Sample Size: Not considering if the sample size is adequate can lead to misleading conclusions.
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