Question 1200096
**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!