Normal Distribution and Z-scores Test Answer Key and Scoring

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

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Looking for a normal distribution and z-scores test answer key? Then this answer key gives the solutions and the scoring guide of the Algebra 2 test on Normal Curves, z-Scores and Study Design. Each answer shows the computation and a scoring guide. For histogram estimates accept an interval within one bar of the values given, and add the points for a total out of 20.

Standards: S-ID.A.4, S-IC.A.1, S-IC.B.3, S-IC.B.4 · Level: Grade 11 Algebra 2

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Question 1 Mean and standard deviation (2 points)

Mean = 40/5 = 8. Squared deviations: 25, 1, 0, 4, 16, sum 46, variance 9.2. σ = √9.2 ≈ 3.033. Mean 8; σ ≈ 3.03.

Scoring: 1 point for the mean, 1 point for σ.

Question 2 z-scores (2 points)

z = (61 − 52)/4.5 = 2. For −0.8: 52 − 0.8(4.5) = 52 − 3.6 = 48.4. z = 2 and x = 48.4.

Scoring: 1 point each.

Question 3 Heights and the Empirical Rule (3 points)

61 and 71 are 2σ from 66: about 95%.
63.5 = 66 − 2.5 is 1σ below: 16% are below. About 16%.
71 is 2σ above; 5% are outside 2σ, so half of that is above. About 2.5%.

The upper tail beyond 71 holds about 2.5%.

Scoring: 1 point per part.

Question 4 Compare swimmers (3 points)

Jo: z = (58.2 − 60)/1.5 = -1.2. Lee: z = (50.5 − 52)/1 = -1.5. Both are below average, but Lee is farther below. Lee did better relative to the group (−1.5 versus −1.2).

Scoring: 1 point for each z-score, 1 point for the conclusion.

Question 5 Classify the study (2 points)

(a) Treatments are randomly assigned: experiment. (b) Data are recorded with no treatment and no sampling from a larger plan: observational study.

Scoring: 1 point each.

Question 6 Margin of error from a histogram (3 points)

Counting about 2 or 3 results in each tail, the middle 95% lies from about 0.33 to 0.67.
The margin of error is half the width: (0.67 − 0.33)/2 = 0.17.

Scoring: 2 points for the interval (1 for each endpoint), 1 point for the margin.

Question 7 Counting students above a score (2 points)

78 is 1σ above the mean, so about 16% scored above. 0.16 × 400 = 64. About 64 students.

Scoring: 1 point for 16%, 1 point for the count.

Question 8 Identify bias (2 points)

Club members are not representative: they may be likelier to have rides home or live nearby. The sample is biased; use a random sample drawn from the whole student list.

Scoring: 1 point for the bias, 1 point for the improvement.

Question 9 Cause and effect (1 point)

A randomized experiment, because random assignment balances all other factors across the groups.

Scoring: 1 point for the choice with a reason.

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