For the model N(41.3, 8.3), find the data points that represent the middle 50% of the data

The notation \( N(41.3, 8.3) \) represents a normal distribution with a mean (\(\mu\)) of 41.3 and a standard deviation (\(\sigma\)) of 8.3.

To find the data points that represent the middle 50% of the data in this normal distribution, we can follow these steps:

1. **Find the z-scores that correspond to the middle 50%**:
   - The middle 50% is centered around the mean, so 25% of the data lies below and 25% lies above this middle region.
   - To find the z-scores, we use the percentiles for the lower and upper bounds. The lower bound will be at the 25th percentile (0.25) and the upper bound at the 75th percentile (0.75).
   - From standard normal distribution tables, the z-scores corresponding to the 25th and 75th percentiles are approximately:
     - \( z_{0.25} = -0.674 \)
     - \( z_{0.75} = 0.674 \)

2. **Convert the z-scores to data points** using the formula:
   \[
   X = \mu + z \sigma
   \]
   where:
   - \( \mu = 41.3 \)
   - \( \sigma = 8.3 \)
   - \( z \) is the z-score

   For the 25th percentile:
   \[
   X_{25} = 41.3 + (-0.674)(8.3) \approx 41.3 - 5.597 = 35.703
   \]

   For the 75th percentile:
   \[
   X_{75} = 41.3 + (0.674)(8.3) \approx 41.3 + 5.597 = 46.897
   \]

Therefore, the data points that represent the middle 50% of the distribution are approximately **35.70** and **46.90**.

Would you like more details on this or have any questions?

Here are 5 follow-up questions to explore:

1. What is the significance of the middle 50% in a normal distribution?
2. How do z-scores help in interpreting data from a normal distribution?
3. What is the empirical rule for normal distributions?
4. How can we find z-scores for any given percentile?
5. How would this change if the standard deviation was smaller or larger?

**Tip**: The middle 50% is also known as the interquartile range (IQR) in descriptive statistics, which provides a sense of the spread of the middle portion of the data.

For the model N(41.3, 8.3), find the data points that represent the middle 50% of the data.

Learn how to calculate the data points representing the middle 50% of a normal distribution with a mean of 41.3 and a standard deviation of 8.3. Using z-scores for the 25th and 75th percentiles, we convert them to approximate data points at 35.70 and 46.90. This solution is perfect for students in Grades 10-12 studying statistics and normal distributions.

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