Suppose that people's heights (in centimeters) are normally distributed, with a mean of 175 and a standard deviation of 6. We find the heights of 80 people.
(a) How many would you expect to be between 167 and 183 cm tall?
(b) How many would you expect to be taller than 170 cm?
my trusty normal distribution calculator tells me that
prob (between 167 and 183) = .8176
prob( > 170) = .7977
multiply each of those by 80
http://davidmlane.com/normal.html
Thank you so much for your response. Just to be clear are you saying I need to multiply .8176 x 80 and then .7977 x 80?
To answer these questions, we can use the properties of the normal distribution and the given mean and standard deviation.
(a) To find the number of people expected to be between 167 and 183 cm tall, we need to calculate the area under the normal curve between these two values.
Step 1: Calculate the z-scores for the lower and upper limits.
The z-score represents the number of standard deviations a data point is from the mean. We can calculate the z-scores using the formula z = (x - μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation.
For the lower limit: z = (167 - 175) / 6 = -1.33
For the upper limit: z = (183 - 175) / 6 = 1.33
Step 2: Use the z-scores to find the area under the normal curve.
We can use a standard normal distribution table or a calculator to find the area corresponding to these z-scores. The area represents the probability of a data point falling between these two values.
Using a standard normal distribution table, we find that the area to the left of z = -1.33 is 0.0918, and the area to the left of z = 1.33 is 0.9082.
Step 3: Calculate the expected number of people.
To find the number of people expected to be between 167 and 183 cm tall, we can multiply the total number of people (80) by the probability found in Step 2.
Expected number = 80 * (0.9082 - 0.0918) = 69.76
Therefore, we would expect approximately 70 people to be between 167 and 183 cm tall.
(b) To find the number of people expected to be taller than 170 cm, we need to calculate the area under the normal curve to the right of 170 cm.
Step 1: Calculate the z-score for the height of 170 cm.
z = (170 - 175) / 6 = -0.83
Step 2: Use the z-score to find the area under the normal curve.
Using a standard normal distribution table or a calculator, we find that the area to the left of z = -0.83 is 0.2033.
Step 3: Calculate the expected number of people.
To find the number of people expected to be taller than 170 cm, we multiply the total number of people (80) by the probability found in Step 2.
Expected number = 80 * (1 - 0.2033) = 63.34
Therefore, we would expect approximately 63 people to be taller than 170 cm.