Algebra 2 The Normal Distribution Test Solutions and Scoring (PDF)

Algebra 2 math test solutions: The Normal Distribution – Maths PDF

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This algebra 2 normal distribution test answers for Algebra 2 covers the normal distribution (Common Core S-ID.A.4), with a free PDF to print.

Use this answer key and scoring guide to grade the test on the normal distribution (S-ID.A.4). Each solution shows the full reasoning, and the scoring note explains how many points to award for correct setup, correct work, and a correct final answer. Partial credit rewards a sound method even when a small arithmetic slip changes the result.

Algebra 2 · Common Core State Standards: S-ID.A.4

Algebra 2 normal distribution test answers: worked solutions

Question 1 – Empirical rule [2 pts]

85 and 115 are within 1 sd.

85 and 115 are 1 standard deviation from 100: 68%. 70 and 130 are 2 standard deviations away: 95%.

About 68% and about 95%

Scoring: 1 point for each percent.

Question 2 – z-score [2 pts]

\(z = \frac{24 – 30}{4} = -1.5\).

z = -1.5: 1.5 standard deviations faster (below) the mean

Scoring: 1 point for the z-score, 1 point for the interpretation.

Question 3 – Compare [3 pts]

Aisha: \(z = \frac{91 – 80}{5} = 2.2\). Ethan: \(z = \frac{96 – 85}{8} = 1.375\).

Aisha

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

Question 4 – Area between [3 pts]

Area between -1.5 and 1.

\(P(Z < 1) = 0.8413\) and \(P(Z < -1.5) = 0.0668\).

Subtract: \(0.8413 – 0.0668 = 0.7745\).

About 77.5%

Scoring: 1 point for each area, 1 point for the difference.

Question 5 – Percent above [2 pts]

\(z = \frac{69 – 64}{2.5} = 2\). \(P(Z > 2) = 0.0228\).

About 2.3%

Scoring: 1 point for the z-score, 1 point for the percent.

Question 6 – Find the value [2 pts]

\(x = 500 + (-0.5)(80) = 460\).

460

Scoring: 1 point for the formula, 1 point for the value.

Question 7 – Count [2 pts]

\(z = \frac{78 – 70}{4} = 2\). About 2.5% of values lie above (0.0228).

\(500 \cdot 0.0228 \approx 11\).

About 11 packages

Scoring: 1 point for the z-score, 1 point for the count.

Question 8 – Percentile [2 pts]

\(x = 60 + 0.84(10) = 68.4\).

About 68.4

Scoring: 1 point for the formula, 1 point for the value.

Question 9 – Find the error [2 pts]

The 95% figure belongs to 2 standard deviations. Within 1 standard deviation lies about 68% of the data.

68%

Scoring: 1 point for the correct percent, 1 point for the explanation.

Total score

Total

20 points

Key facts behind this algebra 2 normal distribution test answers

Key facts

Normal distribution: symmetric, bell-shaped, described by its mean and standard deviation.
68-95-99.7 rule for 1, 2, 3 standard deviations.
z = (x – mean) / sd and x = mean + z times sd.
Area under the curve = proportion of data.
Compare different distributions with z-scores.

How to use this algebra 2 normal distribution test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Algebra 2 chapters: Algebra 2 math.

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