

X is a normally distributed random variable with mean 48 and standard deviation 6.
What is the probability that X is between 42 and 54?

Given that \( X \) is a normally distributed random variable with a mean \( \mu = 48 \) and standard deviation \( \sigma = 6 \), we want to find the probability that \( X \) is between 42 and 54. Mathematically, this is expressed as:

\[
P(42 \leq X \leq 54)
\]

To calculate this, we'll standardize the values to the standard normal distribution using the formula for the z-score:

\[
Z = \frac{X - \mu}{\sigma}
\]

### Step 1: Compute the z-scores for 42 and 54

For \( X = 42 \):

\[
Z_1 = \frac{42 - 48}{6} = \frac{-6}{6} = -1
\]

For \( X = 54 \):

\[
Z_2 = \frac{54 - 48}{6} = \frac{6}{6} = 1
\]

### Step 2: Find the probabilities from the standard normal distribution

From standard normal distribution tables or a calculator:

\[
P(Z \leq 1) \approx 0.8413
\]
\[
P(Z \leq -1) \approx 0.1587
\]

### Step 3: Calculate the probability

The probability that \( X \) is between 42 and 54 is the difference between the two probabilities:

\[
P(42 \leq X \leq 54) = P(Z \leq 1) - P(Z \leq -1) = 0.8413 - 0.1587 = 0.6826
\]

Thus, the probability that \( X \) is between 42 and 54 is approximately **0.6826** or **68.26%**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you find the probability for a different range of values, like \( P(40 \leq X \leq 60) \)?
2. What is the probability that \( X \) is greater than 54?
3. How do you calculate probabilities for non-symmetric ranges in normal distributions?
4. Can you explain how the standard normal table is used in this process?
5. What is the z-score for an \( X \)-value of 60, and what does it tell you?

**Tip:** When using the standard normal distribution table, always ensure to correctly identify the area under the curve corresponding to the z-scores, as this gives you the desired probabilities.

X is a normally distributed random variable with mean 48 and standard deviation 6. What is the probability that X is between 42 and 54?

Find the probability that a normally distributed random variable X with mean 48 and standard deviation 6 is between 42 and 54. This step-by-step guide explains the process of calculating z-scores and using the standard normal distribution to determine probabilities. Suitable for Grades 10-12.

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