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 70 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 175 cm tall?
people
(b) How many would you expect to be taller than 179 cm?
people
Z = (score-mean)/SD = (175-170)/5 = 1
(a) Z = 1, Consulting normal distribution table in you stat text, 34% of 70 = ?
Do the same for (b).
To find the answers to these questions, we need to use the properties of the standard normal distribution and Z-scores.
(a) To find the number of people expected to be between 170 and 175 cm tall, we need to calculate the area under the normal curve between these two heights.
Step 1: Convert the given heights of 170 and 175 cm to Z-scores.
The formula to calculate the Z-score is: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
For a height of 170 cm:
Z1 = (170 - 170) / 5 = 0
For a height of 175 cm:
Z2 = (175 - 170) / 5 = 1
Step 2: Use the standard normal distribution table or a Z-score calculator to find the corresponding cumulative probability for each Z-score.
For Z1 = 0, the cumulative probability is 0.5000.
For Z2 = 1, the cumulative probability is 0.8413.
Step 3: Calculate the area between these two Z-scores.
The difference between the cumulative probabilities gives us the area between the two Z-scores.
Area = Cumulative probability at Z2 - Cumulative probability at Z1
Area = 0.8413 - 0.5000 = 0.3413
Step 4: Convert the area to the number of people.
We know that the total number of people is 70, so we can multiply the area by the total to get the number of people.
Number of people = Area * Total number of people
Number of people = 0.3413 * 70 ≈ 23.86
Rounded to the nearest whole number, we would expect approximately 24 people to be between 170 and 175 cm tall.
(b) To find the number of people expected to be taller than 179 cm, we need to find the area under the normal curve to the right of the Z-score representing 179 cm.
Step 1: Convert the given height of 179 cm to a Z-score.
Z = (X - μ) / σ
Z = (179 - 170) / 5 = 1.8
Step 2: Use the standard normal distribution table or a Z-score calculator to find the cumulative probability for this Z-score.
The cumulative probability at Z = 1.8 is 0.9641.
Step 3: Calculate the area to the right of the Z-score.
The area to the right of the Z-score is equal to 1 minus the cumulative probability.
Area = 1 - Cumulative probability at Z
Area = 1 - 0.9641 = 0.0359
Step 4: Convert the area to the number of people.
Number of people = Area * Total number of people
Number of people = 0.0359 * 70 ≈ 2.51
Rounded to the nearest whole number, we would expect approximately 3 people to be taller than 179 cm.
To find the number of people expected to be between 170 and 175 cm tall, we need to calculate the probability of a randomly selected person falling within this range.
(a) First, we need to standardize the values 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:
z1 = (170 - 170) / 5 = 0
For 175 cm:
z2 = (175 - 170) / 5 = 1
Next, we need to find the cumulative probabilities for these z-scores using the standard normal distribution table:
P(Z < 0) = 0.5000
P(Z < 1) ≈ 0.8413
To find the probability between 170 and 175 cm, we subtract the cumulative probabilities:
P(170 < X < 175) = P(0 < Z < 1) = P(Z < 1) - P(Z < 0) ≈ 0.8413 - 0.5000 ≈ 0.3413
To calculate the number of people expected, we multiply this probability by the total number of people:
Number of people = 0.3413 * 70 ≈ 23.8 ≈ 24
Therefore, we would expect approximately 24 people to be between 170 and 175 cm tall.
(b) To find the number of people expected to be taller than 179 cm, we need to calculate the probability of a randomly selected person being taller than this height.
For 179 cm:
z = (179 - 170) / 5 = 1.8
Using the standard normal distribution table, we can find the probability of a Z-score larger than 1.8:
P(Z > 1.8) ≈ 1 - P(Z < 1.8) ≈ 1 - 0.9641 ≈ 0.0359
To calculate the number of people expected, we multiply this probability by the total number of people:
Number of people = 0.0359 * 70 ≈ 2.5 ≈ 3
Therefore, we would expect approximately 3 people to be taller than 179 cm.