For the model N(142.8, 11.8), find the data point that lies below 73.24% of the data
Z-score



To solve this, we need to find the data point that lies below 73.24% of the data in a normal distribution with mean \( \mu = 142.8 \) and standard deviation \( \sigma = 11.8 \). This is done by finding the corresponding z-score for 73.24%, and then converting that z-score into the actual data point.

### Steps:

1. **Find the z-score** corresponding to 73.24%.
   - Use a standard normal table or a calculator to find the z-score for the cumulative probability of 0.7324. The z-score for this value is approximately **0.61**.

2. **Convert the z-score to the actual data point**:
   The formula for converting a z-score to a data point is:
   \[
   X = \mu + (z \times \sigma)
   \]
   Substituting the values:
   \[
   X = 142.8 + (0.61 \times 11.8)
   \]
   \[
   X = 142.8 + 7.198
   \]
   \[
   X \approx 149.998
   \]

So, the data point that lies below 73.24% of the data is approximately **149.998**.

Would you like more details or clarifications on any part of this?

Here are 5 related questions:
1. What is the z-score formula and how is it used in statistics?
2. How can you interpret the significance of a z-score in data analysis?
3. How would you find a data point for a different percentile in a normal distribution?
4. What are the assumptions underlying a normal distribution?
5. Can z-scores be used for non-normal distributions?

**Tip:** Z-scores are standardized, so they allow comparison across different normal distributions by transforming them to the same scale!

For the model N(142.8, 11.8), find the data point that lies below 73.24% of the data.

Discover how to find the data point that lies below 73.24% of the data in a normal distribution model N(142.8, 11.8). This solution covers the use of z-scores and the formula for converting z-scores to actual data points in statistics. Suitable for high school and college students.

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