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 168 and 182 cm tall?
(b) How many would you expect to be taller than 170 cm?
http://davidmlane.com/hyperstat/z_table.html This is an excellent distribution applet.
Enter the mean, and standard deviation
a) enter between numbers, note the shaded area, and the computation .7566 at the bottom. That means 80*.7566 =60.52 number , which means 60 people between (some add the .52, but it is hard to count .52 folks).
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 between 168 and 182 cm tall, we need to calculate the probability of a random variable falling within this range.
Step 1: Standardize the values
To standardize the values, we subtract the mean from each value and divide by the standard deviation.
For 168 cm:
z1 = (168 - 175) / 6 = -1.1667
For 182 cm:
z2 = (182 - 175) / 6 = 1.1667
Step 2: Find the probability using the standard normal distribution table
Using the standard normal distribution table or a calculator, we can find the probability associated with each z-score.
P(z < -1.1667) = 0.121 (from the table)
P(z < 1.1667) = 0.879 (from the table)
Step 3: Calculate the difference in probabilities
To find the probability between two values (in this case, between -1.1667 and 1.1667), we subtract the probability of the lower value from the probability of the higher value.
P(-1.1667 < z < 1.1667) = P(z < 1.1667) - P(z < -1.1667) = 0.879 - 0.121 = 0.758
Step 4: Calculate the expected number of people
Multiply the probability by the total number of people to get the expected number of people.
Expected number of people = Probability * Total number of people = 0.758 * 80 = 60.64
So, we would expect approximately 61 people to be between 168 and 182 cm tall.
(b) To find the number of people taller than 170 cm, we need to calculate the probability of a random variable being greater than 170 cm.
Step 1: Standardize the value
To standardize the value, we subtract the mean from the value and divide by the standard deviation.
For 170 cm:
z = (170 - 175) / 6 = -0.8333
Step 2: Find the probability using the standard normal distribution table
Using the standard normal distribution table or a calculator, we can find the probability associated with the z-score.
P(z > -0.8333) = 1 - P(z < -0.8333) = 1 - 0.2023 = 0.7977
Step 3: Calculate the expected number of people
Multiply the probability by the total number of people to get the expected number of people.
Expected number of people = Probability * Total number of people = 0.7977 * 80 = 63.82
So, we would expect approximately 64 people to be taller than 170 cm.