Round your answer to the nearest thousandth.
In a population that is normally distributed with mean 10 and standard deviation 17, the bottom 60% of the values are those less than

To find the value below which the bottom 60% of the values lie in a normally distributed population, we need to determine the z-score that corresponds to the 60th percentile and then convert that z-score to the actual value using the given mean and standard deviation.

### Step 1: Find the z-score corresponding to the 60th percentile.
The 60th percentile corresponds to a cumulative probability of 0.60. Using a standard normal distribution table or a calculator, the z-score corresponding to 0.60 is approximately **0.253**.

### Step 2: Convert the z-score to the actual value.
The formula to convert a z-score to an actual value \( X \) is:

\[
X = \mu + z \times \sigma
\]

Where:
- \( \mu = 10 \) (mean)
- \( \sigma = 17 \) (standard deviation)
- \( z = 0.253 \) (z-score corresponding to the 60th percentile)

\[
X = 10 + 0.253 \times 17
\]
\[
X \approx 10 + 4.301 = 14.301
\]

### Step 3: Round to the nearest thousandth.
The value is **14.301**.

Thus, the bottom 60% of the values are those less than **14.301**.

Would you like more details or have any questions about this process?

Here are 5 related questions you might find interesting:
1. How do you find the z-score for a given percentile in a normal distribution?
2. What does the area under the normal curve represent in probability terms?
3. How do changes in the standard deviation affect the spread of a normal distribution?
4. What is the significance of the 50th percentile in a normal distribution?
5. How would you calculate the top 40% of the values in a normal distribution?

**Tip:** Understanding how to use z-scores is crucial for interpreting normal distributions, as it allows you to standardize different datasets for comparison.

Learn how to calculate the value below which the bottom 60% of values lie in a normally distributed population with a mean of 10 and standard deviation of 17. This detailed guide covers the z-score calculation, conversion to actual values, and rounding to the nearest thousandth.

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