The shoe size for men is normally distributed with a mean of 10.5 and a standard deviation of 1.5. What is the probability of selecting a man with a shoe size between 6 and 15?

To solve this problem, we can use the **normal distribution** properties. The shoe sizes for men are normally distributed with a mean \( \mu = 10.5 \) and a standard deviation \( \sigma = 1.5 \). We need to find the probability of selecting a man with a shoe size between 6 and 15, which translates to finding the area under the normal curve between these two values.

The formula to standardize a value from a normal distribution is given by the **z-score** formula:
\[
z = \frac{x - \mu}{\sigma}
\]
where \( x \) is the value we are standardizing, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

### Step 1: Find the z-scores for 6 and 15

For \( x = 6 \):
\[
z_6 = \frac{6 - 10.5}{1.5} = \frac{-4.5}{1.5} = -3
\]

For \( x = 15 \):
\[
z_{15} = \frac{15 - 10.5}{1.5} = \frac{4.5}{1.5} = 3
\]

### Step 2: Use the standard normal distribution table

Now we need to find the probability that the z-score is between -3 and 3. We use the cumulative distribution function (CDF) for the standard normal distribution:

- The probability of a z-score less than 3 is approximately 0.9987 (from standard normal tables).
- The probability of a z-score less than -3 is approximately 0.0013.

### Step 3: Calculate the probability

The probability of selecting a man with a shoe size between 6 and 15 is the difference between these two values:
\[
P(6 < X < 15) = P(Z < 3) - P(Z < -3) = 0.9987 - 0.0013 = 0.9974
\]

Thus, the probability of selecting a man with a shoe size between 6 and 15 is approximately **0.9974**, or **99.74%**.

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Do you need further details or have any questions?

Here are 5 related questions you might find helpful:

1. How do we calculate probabilities for values outside of the range in a normal distribution?
2. What are z-scores, and why are they useful in normal distribution problems?
3. How do we interpret probabilities and areas under the normal curve?
4. What if the shoe sizes were normally distributed with a different mean or standard deviation?
5. How can I use a z-table or software to find exact probabilities for different z-scores?

**Tip**: The standard normal distribution table gives cumulative probabilities from the left tail up to a certain z-score, making it useful for calculating probabilities between ranges.

Find the probability of selecting a man with a shoe size between 6 and 15, using the normal distribution with a mean of 10.5 and a standard deviation of 1.5. Learn how to calculate z-scores and use the cumulative distribution function for accurate probability estimation. Suitable for high school students in Grades 10-12.

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