Question

Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. If we find the heights of 50 people, how many would you expect to be between 160 and 165 cm? Round to the nearest whole number.

Answer 1

To find the number of people expected to be between 160 and 165 cm, we calculate the z-scores for both heights and find the area under the normal distribution curve between these z-scores. The area corresponds to the percentage of people expected to fall within this range. Finally, we multiply this percentage by the total number of people to find the expected number of people between the given heights.

Step-by-step explanation:

To find the number of people expected to be between 160 and 165 cm, we need to calculate the z-scores for both heights and then find the area under the normal distribution curve between these z-scores.

First, we calculate the z-score for 160 cm using the formula:

z = (x - μ) / σ

where x is the height, μ is the mean, and σ is the standard deviation.

Plugging in the values, we get:

z = (160 - 170) / 5 = -2

Next, we calculate the z-score for 165 cm:

z = (165 - 170) / 5 = -1

Now, we can use a z-score table or a calculator to find the area between these z-scores. The area corresponds to the percentage of people expected to fall within this range.

From the z-score table or calculator, we find that the area between -2 and -1 is approximately 0.135. To find the number of people, we multiply this percentage by the total number of people (50).

Therefore, we can expect approximately 7 people to have heights between 160 and 165 cm.