Question 1200096: The mean height of a population of girls aged 15 to 19 years in a certain population was found to be 165 cm with a standard deviation of 15cm. Assuming that the heights are normally distributed, find the heights in centimeters that correspond to the following percentiles:
a. Between the 20th and 50th percentiles.
b. Between the 40th and 60th percentiles.
c. Between the 10th and 90th percentiles.
d. Above the 80th percentile.
e. Below the 10th percentile.
f. Above the 5th percentile.
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! **1. Find Z-scores:**
* **Use a Z-table or calculator (like a TI-84) to find the Z-scores corresponding to the given percentiles.**
**a. Between 20th and 50th percentiles:**
* Z-score for 20th percentile: -0.84
* Z-score for 50th percentile: 0 (mean)
**b. Between 40th and 60th percentiles:**
* Z-score for 40th percentile: -0.25
* Z-score for 60th percentile: 0.25
**c. Between 10th and 90th percentiles:**
* Z-score for 10th percentile: -1.28
* Z-score for 90th percentile: 1.28
**d. Above 80th percentile:**
* Z-score for 80th percentile: 0.84
**e. Below 10th percentile:**
* Z-score for 10th percentile: -1.28
**f. Above 5th percentile:**
* Z-score for 5th percentile: -1.645
**2. Calculate Heights:**
* **Use the formula: X = μ + Zσ**
* where X is the height, μ is the mean, Z is the z-score, and σ is the standard deviation.
**a. Between 20th and 50th percentiles:**
* 20th percentile: X = 165 + (-0.84)(15) = 151.4 cm
* 50th percentile: X = 165 + (0)(15) = 165 cm
* Heights: Between 151.4 cm and 165 cm
**b. Between 40th and 60th percentiles:**
* 40th percentile: X = 165 + (-0.25)(15) = 161.25 cm
* 60th percentile: X = 165 + (0.25)(15) = 168.75 cm
* Heights: Between 161.25 cm and 168.75 cm
**c. Between 10th and 90th percentiles:**
* 10th percentile: X = 165 + (-1.28)(15) = 142.8 cm
* 90th percentile: X = 165 + (1.28)(15) = 187.2 cm
* Heights: Between 142.8 cm and 187.2 cm
**d. Above 80th percentile:**
* 80th percentile: X = 165 + (0.84)(15) = 177.6 cm
* Heights: Above 177.6 cm
**e. Below 10th percentile:**
* 10th percentile: X = 165 + (-1.28)(15) = 142.8 cm
* Heights: Below 142.8 cm
**f. Above 5th percentile:**
* 5th percentile: X = 165 + (-1.645)(15) = 139.325 cm
* Heights: Above 139.325 cm
**Note:** These calculations provide approximate values based on the normal distribution.
Let me know if you'd like to explore any other percentiles or have further questions!
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