Normal Distribution and Z-Scores Flashcards

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?

Answer: 97.72%

z = (69 - 64) / 2.5 = 2.0; P(Z < 2) ≈ 0.9772, or 97.72%.

Which of the following z-scores corresponds to the 84th percentile of a standard normal distribution?

Answer: 1.00

P(Z < 1.00) ≈ 0.8413, which is approximately the 84th percentile.

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?

Answer: 90th

If the area to the left is 0.90, then z = 1.28 corresponds to the 90th percentile.

Two normal distributions have the same mean but different standard deviations. Which statement is true?

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.

A student's AP exam score has a z-score of 2.3. What does this indicate?

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.

SAT Math scores are normally distributed with μ = 530 and σ = 115. About what proportion of students scored between 415 and 645?

Answer: 0.68

415 = 530 - 115 and 645 = 530 + 115, so these are ±1 standard deviation from the mean, giving approximately 68%.

If P(Z < a) = 0.025, what is the value of a for a standard normal distribution?

Answer: -1.96

By symmetry of the normal distribution, P(Z < -1.96) ≈ 0.025, which is also the lower 2.5% tail.