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 170 and 180 cm tall?
_____ people
(b) How many would you expect to be taller than 176 cm?
______ people
One of the best webpages available for this topic
http://davidmlane.com/hyperstat/z_table.html
It replaces your tables at the back of your text.
enter mean of 170
enter sd of 5
a) click on 'between' and enter 170 and 180
multiply that by 60
b) what do you think?
thank you so much Reiny that helps. I got it right! Sorry for late response
To answer these questions, we can use the properties of the standard normal distribution. We will convert the given heights into z-scores using the formula:
z = (x - μ) / σ
where:
- x represents the given value (height in this case)
- μ represents the mean of the distribution (170)
- σ represents the standard deviation of the distribution (5)
(a) To find the number of people expected to be between 170 and 180 cm tall, we need to find the area under the normal curve between these two values. We can calculate the z-scores for both heights:
z1 = (170 - 170) / 5 = 0
z2 = (180 - 170) / 5 = 2
Next, we can use the standard normal distribution table (also known as the z-table) to find the area between these two z-scores. The table provides the proportion of the area under the curve from the left side up to a given z-score. The area between two z-scores can be found by subtracting the smaller z-score area from the larger z-score area.
From the z-table, the area to the left of z = 2 is 0.9772, and the area to the left of z = 0 is 0.5. Therefore, the area between z = 0 and z = 2 is:
0.9772 - 0.5 = 0.4772
Now, we need to find the number of people within this range. Since the total number of people is given as 60, we can multiply the proportion by the total count:
0.4772 * 60 = 28.63
Rounding to the nearest whole number, we expect approximately 29 people to be between 170 and 180 cm tall.
Therefore, the answer is:
(a) 29 people
(b) To find the number of people expected to be taller than 176 cm, we need to find the area to the right of the z-score for 176 cm. We can calculate the z-score as follows:
z = (176 - 170) / 5 = 1.2
From the z-table, the area to the left of z = 1.2 is 0.8849. To find the area to the right, we subtract this value from 1:
1 - 0.8849 = 0.1151
Multiplying this proportion by the total count, we get:
0.1151 * 60 = 6.9
Rounding to the nearest whole number, we expect approximately 7 people to be taller than 176 cm.
Therefore, the answer is:
(b) 7 people
To find the number of people between a certain range or above a certain height in a normal distribution, we can use the concept of z-scores.
A z-score measures how many standard deviations an individual data point is away from the mean. It is calculated as:
z = (x - μ) / σ
Where:
- x is the data point (height in this case)
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
(a) To find the number of people between 170 and 180 cm tall, we need to calculate the z-scores for both heights and then use the standard normal distribution table to find the proportion of values between those z-scores.
For 170 cm:
z1 = (170 - 170) / 5 = 0
For 180 cm:
z2 = (180 - 170) / 5 = 2
Now, using the standard normal distribution table, we can find the proportion between z1 and z2, which represents the area under the curve between those z-scores.
From the table, the area to the left of z = 0 is 0.5000, and for z = 2, it is 0.9772.
Therefore, the proportion between z1 and z2 is 0.9772 - 0.5000 = 0.4772.
To find the number of people in this range, we multiply the proportion by the total number of people:
Number of people = Proportion * Total number of people
= 0.4772 * 60
≈ 28.6
Rounded to the nearest whole number, we would expect around 29 people to be between 170 and 180 cm tall.
(b) To find the number of people taller than 176 cm, we need to find the proportion of values to the right of the corresponding z-score.
For 176 cm:
z = (176 - 170) / 5 = 1.2
Using the standard normal distribution table, we can determine the proportion to the left of z = 1.2, and subtract it from 1 to find the proportion to the right:
Proportion to the right = 1 - Proportion to the left
From the table, the area to the left of z = 1.2 is 0.8849.
Proportion to the right = 1 - 0.8849 = 0.1151.
To find the number of people taller than 176 cm, we multiply the proportion by the total number of people:
Number of people = Proportion * Total number of people
= 0.1151 * 60
≈ 6.9
Rounded to the nearest whole number, we would expect around 7 people to be taller than 176 cm.