Sure, let's work on this step by step.
### Step 1: Understand the Problem
We are given a normal distribution with a mean (μ) of 15.4 and a standard deviation (σ) of 3.9. We need to find the probability that the variable falls within the range 10 to 26, i.e., P(10 ≤ x ≤ 26).
### Step 2: Convert the Raw Scores to Z-Scores
We need to convert the given raw scores (10 and 26) into z-scores. The z-score is calculated using the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
#### Calculating the z-score for the lower bound (10):
[tex]\[ z_{lower} = \frac{10 - 15.4}{3.9} \][/tex]
[tex]\[ z_{lower} \approx -1.3846 \][/tex]
#### Calculating the z-score for the upper bound (26):
[tex]\[ z_{upper} = \frac{26 - 15.4}{3.9} \][/tex]
[tex]\[ z_{upper} \approx 2.7179 \][/tex]
### Step 3: Use the Standard Normal Distribution
Now we need to find the cumulative probability for these z-scores from the standard normal distribution.
#### Finding the cumulative probability for z-lower (-1.3846):
[tex]\[ P(Z \leq -1.3846) \approx 0.0832 \][/tex]
#### Finding the cumulative probability for z-upper (2.7179):
[tex]\[ P(Z \leq 2.7179) \approx 0.9969 \][/tex]
### Step 4: Calculate the Desired Probability
The probability that x falls between 10 and 26 is the difference between the cumulative probabilities for the upper and lower z-scores:
[tex]\[ P(10 \leq x \leq 26) = P(Z \leq 2.7179) - P(Z \leq -1.3846) \][/tex]
[tex]\[ P(10 \leq x \leq 26) \approx 0.9969 - 0.0832 \][/tex]
[tex]\[ P(10 \leq x \leq 26) \approx 0.9136 \][/tex]
### Step 5: Round the Result
Finally, we round this probability to four decimal places.
So, the probability that x falls between 10 and 26 is approximately:
[tex]\[ \boxed{0.9136} \][/tex]
