Suppose that people's heights (in centimeters) are normally distributed, with a mean of 165 and a standard deviation of 6. We find the heights of 80 people.

(a) How many would you expect to be between 158 and 172 cm tall

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion between the two Z scores. Multiply that by 80.

To find the number of people expected to be between 158 and 172 cm tall, we need to calculate the proportion of the population that falls within that range.

First, let's calculate the z-scores for the lower and upper bounds of the range using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.

For the lower bound:
z_lower = (158 - 165) / 6 = -1.17

For the upper bound:
z_upper = (172 - 165) / 6 = 1.17

Next, we need to find the area under the normal distribution curve between these z-scores.

Using a standard normal distribution table or a calculator, we can find that the cumulative probability corresponding to a z-score of -1.17 is approximately 0.121, and the cumulative probability corresponding to a z-score of 1.17 is approximately 0.879.

So, the proportion of the population between these z-scores is:
P(158 ≤ x ≤ 172) = P(z_lower ≤ Z ≤ z_upper) = P(-1.17 ≤ Z ≤ 1.17) = 0.879 - 0.121 = 0.758

To find the expected number of people in this range, we multiply the proportion by the total number of people (80):
Expected number = 0.758 * 80 ≈ 60.67

Therefore, we would expect approximately 61 people to have heights between 158 and 172 cm.