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 40 people.
(a) How many would you expect to be between 158 and 172 cm tall?
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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 100 people.
a) How many would you expect to be between 158 and 172 cm tall?
b) How many would you expect to be taller than 160 cm tall?
To find out how many people you would expect to be between 158 and 172 cm tall, you need to calculate the probability of a height falling within that range.
First, you need to standardize the range using the z-score formula:
z = (x - μ) / σ
where x is the value you want to standardize (158 and 172 in this case), μ is the mean (165), and σ is the standard deviation (6).
For 158 cm:
z1 = (158 - 165) / 6 = -1.1667
For 172 cm:
z2 = (172 - 165) / 6 = 1.1667
Now, you can use a standard normal distribution table (also known as the Z-table) or a statistical software to find the probability corresponding to these z-scores. Alternatively, you can use a calculator or an online calculator that provides the probability directly.
The probability of being between -1.1667 and 1.1667 can be found using the cumulative distribution function (CDF).
P(-1.1667 ≤ Z ≤ 1.1667) = P(Z ≤ 1.1667) - P(Z ≤ -1.1667)
Using the Z-table or a calculator, you can look up the probabilities. Assuming a standard normal distribution, the probability is approximately 0.675 for each tail.
P(-1.1667 ≤ Z ≤ 1.1667) = 0.675 - 0.325 = 0.35
Finally, you multiply this probability by the total number of people (40) to find the expected number of people between 158 and 172 cm tall.
Expected number = Probability * Total number of people = 0.35 * 40 = 14
Therefore, you would expect approximately 14 people to be between 158 and 172 cm tall.