AP Stats Normal Distribution and Z-Scores 2 — Questions and Answers
Question 1: Heights of adult women are approximately normal with mean 64 inches and standard deviation 2.5 inches. What percentage of women are shorter than 69 inches?
- 84.13%
- 97.72% (Correct answer)
- 95.44%
- 68.26%
Correct answer: 97.72%
z = (69 - 64) / 2.5 = 2.0; P(Z < 2) ≈ 0.9772, or 97.72%.
Question 2: Which of the following z-scores corresponds to the 84th percentile of a standard normal distribution?
- -1.00
- 0.84
- 1.00 (Correct answer)
- 1.28
Correct answer: 1.00
P(Z < 1.00) ≈ 0.8413, which is approximately the 84th percentile.
Question 3: A normal distribution has the property that the area to the left of z = 1.28 is approximately 0.90. What percentile does z = 1.28 represent?
- 12.8th
- 80th
- 90th (Correct answer)
- 95th
Correct answer: 90th
If the area to the left is 0.90, then z = 1.28 corresponds to the 90th percentile.
Question 4: Two normal distributions have the same mean but different standard deviations. Which statement is true?
- The distribution with the larger standard deviation is taller and narrower.
- The distribution with the smaller standard deviation is taller and narrower. (Correct answer)
- Both distributions have the same height at the mean.
- The standard deviation does not affect the shape of the distribution.
Correct answer: The distribution with the smaller standard deviation is taller and narrower.
A smaller standard deviation means data is more concentrated near the mean, producing a taller and narrower bell curve.
Question 5: A student's AP exam score has a z-score of 2.3. What does this indicate?
- The score is 2.3 points above average.
- The score is 2.3 standard deviations above the mean. (Correct answer)
- The student is in the 23rd percentile.
- The score is in the top 2.3% of all scores.
Correct answer: The score is 2.3 standard deviations above the mean.
A z-score measures the number of standard deviations a value is from the mean; z = 2.3 means 2.3 standard deviations above.
Question 6: SAT Math scores are normally distributed with μ = 530 and σ = 115. About what proportion of students scored between 415 and 645?
- 0.50
- 0.68 (Correct answer)
- 0.95
- 0.997
Correct answer: 0.68
415 = 530 - 115 and 645 = 530 + 115, so these are ±1 standard deviation from the mean, giving approximately 68%.
Question 7: If P(Z < a) = 0.025, what is the value of a for a standard normal distribution?
- 1.96
- -1.96 (Correct answer)
- 2.33
- -2.33
Correct answer: -1.96
By symmetry of the normal distribution, P(Z < -1.96) ≈ 0.025, which is also the lower 2.5% tail.
AP Stats Normal Distribution and Z-Scores 2
Heights of adult women are approximately normal with mean 64 inches and standard deviation 2.5 inches.
What percentage of women are shorter than 69 inches?