Normal Distribution and Z-Scores Flashcards

The weight of a particular brand of cereal box is normally distributed with mean 16.1 oz and standard deviation 0.3 oz. A box is selected at random. What is the probability it weighs less than 16.0 oz?

Answer: 0.3707

z = (16.0 - 16.1) / 0.3 ≈ -0.33; P(Z < -0.33) ≈ 0.3707.

If X ~ N(100, 15²), what is P(85 < X < 130)?

Answer: 0.8186

z₁ = (85-100)/15 = -1, z₂ = (130-100)/15 = 2; P(-1 < Z < 2) = 0.9772 - 0.1587 = 0.8185 ≈ 0.8186.

To use normal probability calculations, which condition about the data is most important to verify?

Answer: The data are approximately normally distributed or the sample is large.

Normal probability calculations require that the underlying population is approximately normal, or that the Central Limit Theorem applies for large samples.

A machine fills bottles with a mean of 500 mL and standard deviation 5 mL (normal distribution). What minimum volume would be exceeded by only 5% of bottles?

Answer: 508.2 mL

P(X > x) = 0.05 means x = 500 + 1.645(5) = 500 + 8.225 ≈ 508.2 mL.

A normal distribution is symmetric about its mean. If P(X > 70) = 0.2, what is P(X < 70)?

Answer: 0.8

Since the total area under the curve equals 1, P(X 70) = 1 - 0.2 = 0.8.

Which of the following correctly uses the normal distribution to find P(X = 72) when X is continuous?

Answer: P(X = 72) = 0, since X is a continuous random variable.

For any continuous distribution, the probability of any single exact value is 0; probabilities are only defined over intervals.

A professor finds that final exam scores are normally distributed with mean 78 and standard deviation 9. She wants to give A grades to the top 15% of students. What is the minimum score for an A?

Answer: 87.3

The 85th percentile has z ≈ 1.04; x = 78 + 1.04(9) = 78 + 9.36 ≈ 87.4, closest to 87.3.