Data and Statistical Literacy Quiz | Applied Averages, Percentages, Sampling and Test Accuracy

The middle of the five ordered values is 33, and the median is not greatly shifted by the extreme value of 500. The mean is (30 + 32 + 33 + 35 + 500) ÷ 5 = 126, far from most employees' salaries. The median suits the stated goal, although the mean may be meaningful for a different purpose, such as spreading total payroll evenly across employees.

The difference between the two rates is 50% − 40% = 10 percentage points. The relative increase divides that 10-point difference by the original 40%: 10 ÷ 40 = 0.25, or 25%. A change in percentage points is not the same as a percentage change relative to the original value.

The absolute risk reduction is 2% − 1% = 1 percentage point. The relative risk reduction divides that difference by the control group's 2%: 1 ÷ 2 = 50%. Reporting only a 50% reduction hides the original risk level, so showing both absolute and relative values makes the scale of the effect easier to judge.

A large response count does not guarantee representativeness when the participation process is biased. Here, customers with strong complaints are more likely to respond, creating possible self-selection bias. A sample of 100,000 may reduce random sampling error, but it does not automatically repair a biased recruitment method.

Random assignment helps prevent participants' existing characteristics from being systematically concentrated in one group, strengthening causal comparison within the volunteer sample. The participants, however, volunteered after seeing an advertisement and were not randomly sampled from all students. Whether the result generalizes to all students is therefore a separate question.

For sensitivity, the denominator is the 100 people who truly have the condition: 90 ÷ 100 = 90%. For specificity, the denominator is the 900 people who do not have it: 810 ÷ 900 = 90%. Using all 180 positive test results as the denominator calculates positive predictive value, not sensitivity.

Of the 100 people with the condition, 90%—or 90 people—are true positives. Of the 9,900 people without it, 10%—or 990 people—are false positives. There are therefore 1,080 positive results, and the positive predictive value is 90 ÷ 1,080 ≈ 8.3%. Even with 90% sensitivity and specificity, false positives can outnumber true positives when the condition is rare.

In frequentist inference, the population mean is a fixed unknown value, while the interval changes from sample to sample. A 95% confidence procedure means that about 95% of intervals produced by repeated use of the same sampling and calculation method would contain the mean. It does not assign 95% probability to the fixed mean being inside this already calculated interval.

With other conditions held constant, the standard error of a mean is inversely proportional to the square root of sample size n. Since 1,600 is four times 400 and its square root is twice as large, B's margin of error is about one-half of A's. Increasing sample size does not automatically remove systematic bias from self-selection or measurement methods.

A's rate is 1,000 ÷ 1,000,000 × 100,000 = 100 per 100,000 people. B's is 300 ÷ 100,000 × 100,000 = 300 per 100,000. A has more events, but B's population-based rate is three times A's. If factors such as age distributions differ greatly, adjusted rates may also need to be considered alongside crude rates.