Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. We find the heights of 60 people. (You may need to use the standard normal distribution table. Round your answers to the nearest whole number.)

How many would you expect to be between 170 and 175 cm tall?

How many would you expect to be taller than 178 cm?

Z = (score-mean)/SD

Consult your table with the Z score. Multiply by 60.

To find out how many people are expected to be between 170 and 175 cm tall, we need to calculate the area under the normal curve between those two values. We can use the standard normal distribution table for this.

First, we need to standardize the values of 170 and 175 cm using the formula: z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation.

For 170 cm:
z = (170 - 170) / 5 = 0

For 175 cm:
z = (175 - 170) / 5 = 1

Next, we need to find the area under the normal curve between z = 0 and z = 1. We can use the standard normal distribution table to find this probability.

From the table, the area to the left of z = 1 is approximately 0.8413, and the area to the left of z = 0 is 0.5000. So, the area between z = 0 and z = 1 is:

0.8413 - 0.5000 = 0.3413

To find the number of people expected to be between 170 and 175 cm tall, we multiply this probability by the total number of people (60):

0.3413 * 60 = 20.478

Rounding to the nearest whole number, we would expect approximately 20 people to be between 170 and 175 cm tall.

To find out how many people are expected to be taller than 178 cm, we need to calculate the area under the normal curve to the right of 178 cm. Again, we will standardize the value of 178 cm using the same formula:

z = (178 - 170) / 5 = 1.6

Next, we need to find the area to the right of z = 1.6 using the standard normal distribution table.

The area to the left of z = 1.6 is approximately 0.9452. However, we want the area to the right of 1.6, so we subtract this value from 1:

1 - 0.9452 = 0.0548

To find the number of people expected to be taller than 178 cm, we multiply this probability by the total number of people (60):

0.0548 * 60 = 3.288

Rounding to the nearest whole number, we would expect approximately 3 people to be taller than 178 cm.