Algebra 2 The Normal Distribution Worksheet Solutions (PDF)
This algebra 2 normal distribution answers for Algebra 2 covers the normal distribution (Common Core S-ID.A.4), with a free PDF to print.
Every worksheet problem on the normal distribution is solved here, one step at a time. Compare your method as well as your final answer, because the steps show where confusing standard deviations with percents can go wrong. Final answers appear in bold so you can find them quickly when you only need a check.
Algebra 2 · Common Core State Standards: S-ID.A.4
Algebra 2 normal distribution answers: exercises 1 to 10
Exercise 1 – Within one standard deviation
\(66 – 3 = 63\) and \(66 + 3 = 69\): within 1 standard deviation.
About 68%
Exercise 2 – Within two standard deviations
\(66 \pm 2(3)\) gives 60 and 72.
About 95%
Exercise 3 – Tail percent
\(75 = 66 + 3(3)\), so 75 is 3 standard deviations above the mean. 99.7% are within 3, leaving \(100\% – 99.7\% = 0.3\%\) in the two tails.
By symmetry, half of 0.3% is in the upper tail.
About 0.15%
Exercise 4 – Compute a z-score
\(z = \frac{84 – 72}{6} = 2\).
z = 2
Exercise 5 – Negative z-score
\(z = \frac{55 – 70}{10} = -1.5\).
z = -1.5: 1.5 standard deviations below the mean
Exercise 6 – From z to a value
\(x = \mu + z\sigma = 500 + 1.2(100) = 620\).
620
Exercise 7 – Compare two tests
Maya: \(z = \frac{88 – 80}{4} = 2\). Jamal: \(z = \frac{85 – 72}{6} \approx 2.17\).
Jamal did better (z about 2.17 compared with 2)
Exercise 8 – Area below z = 1
By a standard normal table or calculator, \(P(Z < 1) \approx 0.8413\).
About 84.1%
Exercise 9 – Area above z = 1.5
\(P(Z > 1.5) = 1 – P(Z < 1.5) = 1 - 0.9332 = 0.0668\).
About 6.68%
Exercise 10 – Area between
\(P(-1 < Z < 2) = 0.9772 - 0.1587 = 0.8186\).
About 81.9%
Exercises 11 and up
Exercise 11 – Percent below a score
\(z = \frac{1250 – 1050}{200} = 1\). \(P(Z < 1) \approx 0.8413\).
About 84.1%
Exercise 12 – Percent above a score
\(z = \frac{1400 – 1050}{200} = 1.75\). \(P(Z > 1.75) = 1 – 0.9599 = 0.0401\).
About 4.0%
Exercise 13 – Percent between two scores
\(z = -0.75\) and \(z = 0.75\). \(P(-0.75 < Z < 0.75) = 0.7734 - 0.2266 = 0.5468\).
About 54.7%
Exercise 14 – Percentile
\(x = 1050 + 1.28(200) = 1306\).
About 1,306
Exercise 15 – Light bulbs
\(z = \frac{700 – 800}{50} = -2\). About 2.5% of values are below \(z = -2\): \(P(Z < -2) \approx 0.0228\).
About 2.3%
Exercise 16 – How many bulbs?
\(z = 1\). \(P(Z > 1) \approx 0.1587\).
\(2000 \cdot 0.1587 \approx 317\).
About 317 bulbs
Exercise 17 – Bottle filling
\(16.0 \pm 3(0.1)\) gives 15.7 and 16.3, so 99.7% pass.
Rejected: \(0.3\%\). \(4000 \cdot 0.003 = 12\).
About 0.3%, which is about 12 bottles
Exercise 18 – Mean and standard deviation
Mean: \(\frac{4 + 6 + 8 + 10 + 12}{5} = 8\).
Squared deviations: \(16, 4, 0, 4, 16\), total 40, variance \(\frac{40}{5} = 8\), so \(\sigma = \sqrt{8} \approx 2.83\).
\(z = \frac{12 – 8}{2.83} \approx 1.41\).
Mean 8; standard deviation about 2.83; z about 1.41
Exercise 19 – Is a normal model reasonable?
73 and 83 are within 1 standard deviation of 78. \(\frac{17}{25} = 0.68\).
That matches the 68% predicted by the empirical rule.
Yes, 68% is consistent with a normal model
Exercise 20 – Find the error
A z-score counts standard deviations, not percent. A z-score of 2 means the value is 2 standard deviations above the mean, which is \(\mu + 2\sigma\).
z = 2 means 2 standard deviations above the mean
Key facts behind this algebra 2 normal distribution answers
- 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 answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
Download the algebra 2 normal distribution answers as a PDF with the button below and keep it with your notes.
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