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.)
(a) How many would you expect to be between 165 and 175 cm tall?
(b) How many would you expect to be taller than 167 cm?
See previous post.
Multiply proportion found by 60.
To find the answers to these questions, we will use the properties of the normal distribution. Specifically, we will convert the given values to z-scores, which represent the number of standard deviations away from the mean a particular value is.
(a) To calculate the number of people expected to be between 165 and 175 cm tall, we need to find the area under the normal distribution curve between these values.
Step 1: Convert the values to z-scores using the formula: z = (x - μ) / σ
For the lower bound, 165 cm:
z1 = (165 - 170) / 5 = -1
For the upper bound, 175 cm:
z2 = (175 - 170) / 5 = 1
Step 2: Use a standard normal distribution table, also called a z-table, to find the area between these two z-scores. The table provides the area under the curve to the left of a given z-score.
From the z-table, the area to the left of z = 1 is approximately 0.8413, and the area to the left of z = -1 is approximately 0.1587. Therefore, the area between z = -1 and z = 1 is equal to 0.8413 - 0.1587 = 0.6826.
Step 3: Multiply the calculated area by the total number of people (60) to find the number of people between 165 and 175 cm tall:
Number of people = Area * Total number of people = 0.6826 * 60 = 41 (rounded to the nearest whole number)
So, we would expect approximately 41 people to be between 165 and 175 cm tall.
(b) To calculate the number of people expected to be taller than 167 cm, we need to find the area to the right of this value.
Step 1: Convert the value to a z-score:
z = (167 - 170) / 5 = -0.6
Step 2: Use the z-table to find the area to the left of z = -0.6.
From the table, the area to the left of z = -0.6 is approximately 0.2743.
Step 3: Subtract the area to the left of z = -0.6 from 1 to find the area to the right:
Area to the right = 1 - 0.2743 = 0.7257
Step 4: Multiply the calculated area by the total number of people (60):
Number of people = Area * Total number of people = 0.7257 * 60 = 44 (rounded to the nearest whole number)
Therefore, we would expect approximately 44 people to be taller than 167 cm.