Recycling Survey and a Newspaper Claim
A town’s environmental committee surveyed 240 residents about whether they recycle regularly. The results are split by age group. Among the 90 residents under 30, 54 said they recycle regularly and 36 said they do not. Among the 150 residents aged 30 and over, 105 said they recycle regularly and 45 said they do not. A local newspaper headline reads: 'Recycling is universal: about two-thirds of all residents recycle regularly, and age makes no difference.' The committee suspects the headline overstates the case. They want a probability-based check before writing a rebuttal. The survey is treated as a random sample of town residents, and the committee will use a 5% significance level only as an informal benchmark, not a formal test. No resident is counted twice.
1. Calculate the probability that a randomly selected respondent recycles regularly, and the conditional probabilities given each age group. Use these to evaluate the 'age makes no difference' part of the headline.
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Total recycling regularly = 54+105=159 out of 240, so P(regular)=159/240=0.6625≈66.25%. Conditional probabilities: P(regular | under 30)=54/90=0.60; P(regular | 30 and over)=105/150=0.70. The two age groups differ by 0.10 (10 percentage points), so the sample does not support the statement that age makes no difference. The overall two-thirds figure is close to 66.25%, but it hides that difference.
2. Determine whether the events 'under 30' and 'recycles regularly' are independent using a numerical check. Show the comparison you use.
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Events are independent if P(regular | under 30)=P(regular). Here 0.60 ≠ 0.6625, so the events are not independent in the sample. Equivalently, the expected number of under-30 regular recyclers under independence would be (90×159)/240=59.625, but the observed count is 54. The observed count is lower than expected. This numerical check supports an association between age group and recycling, though a formal test would be needed to generalize beyond the sample.
3. The newspaper says 'about two-thirds of all residents recycle regularly.' Is that statement accurate? Does the survey justify the stronger claim that age makes no difference? Write a short rebuttal using the numbers.
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The statement 'about two-thirds of all residents recycle regularly' is a reasonable summary of the sample estimate 159/240=66.25%, provided it is treated as an estimate with sampling variability. The stronger claim that age makes no difference is not justified: under-30 recycling is 60%, while 30-and-over recycling is 70%. A short rebuttal could say: 'About two-thirds of respondents recycle regularly, but regular recycling is 10 percentage points higher among residents aged 30 and over (70% vs 60%). The survey therefore suggests an age difference rather than proving none.' The headline should be corrected to avoid claiming no difference.