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 80 people.
(a) How many would you expect to be between 170 and 175 cm tall?
(b) How many would you expect to be taller than 179 cm
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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 expected to be between 170 and 175 cm tall, we need to calculate the probability of a person having a height within this range.
Step 1: Convert the given values to z-scores using the formula:
z = (x - μ) / σ
Where:
x = the given value (either 170 or 175 in this case),
μ = the mean (170),
σ = the standard deviation (5).
For 170 cm:
z1 = (170 - 170) / 5 = 0
For 175 cm:
z2 = (175 - 170) / 5 = 1
Step 2: Look up the z-scores in the standard normal distribution table (also known as the Z-table) to find the corresponding probabilities.
From the table, the probability of a z-score of 0 is 0.5000, and the probability of a z-score of 1 is 0.8413.
Step 3: Calculate the difference between the two probabilities to find the probability of a person having a height between 170 and 175 cm.
P(170 ≤ x ≤ 175) = P(z ≤ 1) - P(z ≤ 0) = 0.8413 - 0.5000 = 0.3413
Step 4: Multiply the probability by the total number of people (80) to find the expected number of people within the given height range.
Expected number = Probability * Total number of people
Expected number = 0.3413 * 80
Expected number ≈ 27.30
To answer (a), we would expect approximately 27 people to be between 170 and 175 cm tall.
(b) To determine the number of people expected to be taller than 179 cm, we need to calculate the probability of a person having a height greater than 179 cm.
Step 1: Calculate the z-score for 179 cm using the same formula as in (a).
z = (179 - 170) / 5 = 1.8
Step 2: Look up the z-score in the standard normal distribution table to find the corresponding probability.
From the table, the probability of a z-score of 1.8 is 0.9641.
Step 3: Calculate the probability of a person having a height taller than 179 cm.
P(x > 179) = 1 - P(z ≤ 1.8) = 1 - 0.9641 = 0.0359
Step 4: Multiply the probability by the total number of people (80) to find the expected number of people taller than 179 cm.
Expected number = Probability * Total number of people
Expected number = 0.0359 * 80
Expected number ≈ 2.87
To answer (b), we would expect approximately 3 people to be taller than 179 cm.