Examine the following sample data and estimate the variance of the population from which it was drawn:

22, 18, 17, 20, 21

Question

Examine the following sample data and estimate the variance of the population from which it was drawn:

22, 18, 17, 20, 21

BlackTom AI · Accepted Answer

To assess the variance from the data set 22, 18, 17, 20, 21, I will compute the mean first and then the squared deviations.
The numbers are: 22, 18, 17, 20, 21.
Compute the mean: (22 + 18 + 17 + 20 + 21) / 5 = 98 / 5 = 19.6.
Next, find each squared deviation from the mean: (22 − 19.6)² = 2.4² = 5.76; (18 − 19.6)² = (−1.6)² = 2.56; (17 − 19.6)² = (−2.6)² = 6.76; (20 − 19.6)² = 0.4² = 0.16; (21 − 19.6)² = 1.4² = 1.96.
Sum of squared deviations = 5.76 + 2.56 + 6.76 + 0.16 + 1.96 = 17.20.
Now, there is an important distinction between population variance and sample variance:
- Population variance uses division by n (here 5), giving 17.20 / 5 = 3.44.
- Sample variance uses division by n−1 (here 4), giving 17.20 / 4 = 4.30.
The options provided are 1.9, 2.1, 5.0, and 4.3.
Option 4.3 corresponds to the sample variance (since 17.20 ÷ 4 = 4.30).
If the intent was to estimate the population variance from the sample, the conventional approach would report 3.44, but that value is not among the choices, so the test item aligns with the sample variance.

Analyzing each option:
- Option 1: 1.9 would require a sum of squared deviations around 7.6 (since 1.9 × 4 ≈ 7.6), which does not match the actual sum 17.20; thus this is inconsistent with the data.
- Option 2: 2.1 implies a total squared deviation of about 8.4 if using n−1, which again is far from the computed 17.20; incorrect for both population and sample interpretations.
- Option 3: 5.0 would correspond to a total squared deviation of 20 if using n−1, or 25 if using n; both are not equal to 17.20, making it inaccurate given the dataset.
- Option 4: 4.3 matches the calculated sample variance (17.20 ÷ 4 = 4.30); this is the value obtained when using the n−1 denominator, which aligns with typical sample-variance computation for estimating the population variance from a small sample.

ISE 4120 SP2025 (10663) EXAM: Unit 1

Examine the following sample data and estimate the variance of the population from which it was drawn: 22, 18, 17, 20, 21

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