SOLUTION: 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 40 people. (You may need to use th

Question 1170059: 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 40 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 165 and 175 cm tall?
(B) How many would you expect to be taller than 168 cm?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's solve this problem step-by-step.
**Given Information:**
* Population mean (μ) = 170 cm
* Population standard deviation (σ) = 5 cm
* Sample size (n) = 40
**A. Expected Number Between 165 and 175 cm:**
1. **Calculate Z-scores:**
* For 165 cm: z₁ = (165 - 170) / 5 = -5 / 5 = -1
* For 175 cm: z₂ = (175 - 170) / 5 = 5 / 5 = 1
2. **Find Probabilities:**
* P(Z < 1) = 0.8413 (from the standard normal distribution table)
* P(Z < -1) = 0.1587 (from the standard normal distribution table)
3. **Calculate the Probability Between 165 and 175 cm:**
* P(-1 < Z < 1) = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826
4. **Calculate the Expected Number:**
* Expected number = P(-1 < Z < 1) * n = 0.6826 * 40 = 27.304
5. **Round to the Nearest Whole Number:**
* Expected number ≈ 27 people
**(A) Approximately 27 people would be expected to be between 165 and 175 cm tall.**
**B. Expected Number Taller Than 168 cm:**
1. **Calculate Z-score:**
* For 168 cm: z = (168 - 170) / 5 = -2 / 5 = -0.4
2. **Find Probability:**
* P(Z > -0.4) = 1 - P(Z < -0.4)
* P(Z < -0.4) = 0.3446 (from the standard normal distribution table)
* P(Z > -0.4) = 1 - 0.3446 = 0.6554
3. **Calculate the Expected Number:**
* Expected number = P(Z > -0.4) * n = 0.6554 * 40 = 26.216
4. **Round to the Nearest Whole Number:**
* Expected number ≈ 26 people
**(B) Approximately 26 people would be expected to be taller than 168 cm.**

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