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 158 and 172 cm tall?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion between the Z scores. Multiply that by 80.
To find the number of people expected to be between 158 and 172 cm tall, we need to calculate the proportion of people within that height range and then multiply it by the total number of people.
First, let's calculate the z-scores for both ends of the height range:
For 158 cm: z = (158 - 165) / 6
For 172 cm: z = (172 - 165) / 6
Next, we can use a standard normal distribution table or a calculator with a cumulative distribution function (CDF) to find the area under the normal curve between these two z-scores. This area represents the proportion of people whose heights fall within the range of 158 to 172 cm.
Using a standard normal table or calculator, we find that the proportion of people between these two z-scores is approximately 0.6827.
Finally, we multiply this proportion by the total number of people (80) to get the expected number of people within the given height range:
Expected number of people = Proportion * Total number of people
Expected number of people = 0.6827 * 80
Expected number of people ≈ 54.62
Therefore, we would expect approximately 54 people out of the 80 to be between 158 and 172 cm tall.