Question 1168059
Here's how to find the probability $P(80 < X < 86)$, including a completely labeled sketch of the normal distribution:

**1. Understand the Problem:**

We have a normally distributed random variable $X$ representing test scores with a mean ($\mu$) of 84.2 and a standard deviation ($\sigma$) of 12.8. We want to find the probability that a randomly selected student's score falls between 80 and 86.

**2. Convert the Scores to Z-Scores:**

To find the probability using the standard normal distribution table or calculator, we need to convert the given scores (80 and 86) into z-scores using the formula:

$z = \frac{X - \mu}{\sigma}$

* For $X = 80$:
    $z_1 = \frac{80 - 84.2}{12.8} = \frac{-4.2}{12.8} \approx -0.3281$

* For $X = 86$:
    $z_2 = \frac{86 - 84.2}{12.8} = \frac{1.8}{12.8} \approx 0.1406$

So, the probability $P(80 < X < 86)$ is equivalent to $P(-0.3281 < Z < 0.1406)$, where $Z$ is the standard normal random variable.

**3. Find the Probabilities Using the Standard Normal Distribution:**

We need to find the area under the standard normal curve between $z_1 = -0.3281$ and $z_2 = 0.1406$. We can do this by finding the cumulative probabilities $P(Z < 0.1406)$ and $P(Z < -0.3281)$ and then subtracting the smaller from the larger.

* $P(Z < 0.1406) \approx 0.5557$ (using a standard normal distribution table or calculator)
* $P(Z < -0.3281) \approx 0.3713$ (using a standard normal distribution table or calculator)

Now, subtract the probabilities:

$P(-0.3281 < Z < 0.1406) = P(Z < 0.1406) - P(Z < -0.3281) \approx 0.5557 - 0.3713 = 0.1844$

**4. Sketch of the Distribution:**

```
                      Normal Distribution
                           / \
                          /   \
                         /     \
                        /       \
                       /         \
                      /           \
                     /             \
        ----------------------------------------------------
       -3s    -2s    -1s     μ     +1s    +2s    +3s   Z-scores

        -3(12.8) -2(12.8) -1(12.8) 84.2 +1(12.8) +2(12.8) +3(12.8) Raw Scores
        (-38.4)  (-25.6)  (-12.8)      (12.8)   (25.6)   (38.4)  Deviations from Mean

        55.8     68.6     71.4    84.2   97.0    109.8   122.6  Approximate Raw Scores

                                 |       |
                                 80      86
                                 ^       ^
                                 |       |
                             z=-0.33  z=0.14

        <----------------------- Shaded Area ----------------------->
```

**Labeling the Sketch:**

* **Horizontal Axis:** Labeled as "Test Scores (X)" and also showing the corresponding "Z-scores" below.
* **Vertical Axis:** Represents the probability density (not explicitly labeled with values, but the curve shows the shape of the distribution).
* **Curve:** A bell-shaped curve representing the normal distribution.
* **Mean (μ):** A vertical line drawn at the mean score of 84.2, labeled with "μ = 84.2".
* **Standard Deviations (s):** Approximate locations of 1, 2, and 3 standard deviations away from the mean are indicated on the raw score axis.
* **Scores of Interest:** Vertical lines are drawn at the scores 80 and 86 on the horizontal axis.
* **Corresponding Z-scores:** The z-scores calculated for 80 (-0.33 approximately) and 86 (0.14 approximately) are indicated below the respective raw scores.
* **Shaded Area:** The area under the curve between the vertical lines at 80 and 86 (or their corresponding z-scores) is shaded. This shaded area represents the probability $P(80 < X < 86)$.

**Answer:**

The probability $P(80 < X < 86) \approx \boxed{0.1844}$.