Suppose that people's heights (in centimeters) are normally distributed, with a mean of 165 and a standard deviation of 6. We find the heights of 80 people.
(a) How many would you expect to be between 159 and 171 cm tall?
To determine how many people we would expect to be between 159 and 171 cm tall, we need to calculate the z-scores for these heights using the formula:
z = (x - μ) / σ
where:
x = height value
μ = mean (165 cm)
σ = standard deviation (6 cm)
For 159 cm:
z1 = (159 - 165) / 6 = -1
For 171 cm:
z2 = (171 - 165) / 6 = 1
Next, we need to find the cumulative probability corresponding to these z-scores using a standard normal distribution table or calculator. The cumulative probability represents the area under the standard normal curve up to that point.
From the standard normal distribution table, the cumulative probability for z = -1 is approximately 0.1587, and the cumulative probability for z = 1 is approximately 0.8413.
To find the probability between these two heights, we subtract the cumulative probability for z = -1 from the cumulative probability for z = 1:
P(159 ≤ x ≤ 171) = P(z ≤ 1) - P(z ≤ -1)
= 0.8413 - 0.1587
= 0.6826
This means that approximately 68.26% of the 80 people would be expected to have heights between 159 and 171 cm tall.
To calculate the number of people, we multiply the probability by the total number of people:
Expected number = probability * total number
= 0.6826 * 80
≈ 54.61
Therefore, we would expect approximately 55 people to be between 159 and 171 cm tall.
To find the number of people you would expect to be between 159 and 171 cm tall, we need to calculate the z-scores for these heights and use the standard normal distribution table.
The formula for calculating the z-score is given by:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the individual height
- μ is the mean
- σ is the standard deviation
For the lower boundary of 159 cm:
z1 = (159 - 165) / 6 = -1.0
For the upper boundary of 171 cm:
z2 = (171 - 165) / 6 = 1.0
Now, we can use the standard normal distribution table or a calculator to find the area between these z-scores.
Using the standard normal distribution table, we can look up the area corresponding to a z-score of -1.0, which is 0.1587, and the area corresponding to a z-score of 1.0, which is 0.8413.
To find the area between these two z-scores, we subtract the smaller area from the larger area:
0.8413 - 0.1587 = 0.6826
This means that approximately 68.26% of the population falls between 159 and 171 cm tall.
To find the number of people from a sample of 80 that fall within this range, we multiply the percentage by the sample size:
0.6826 * 80 = 54.6
Therefore, you would expect approximately 55 people to be between 159 and 171 cm tall in a sample of 80 people.