Stats Medic - Intro Stats - Normal Distribution Calculations ... - Free Printable

It looks like you've uploaded an image containing a statistics problem involving the normal distribution, specifically focusing on z-scores and percentiles. Let's break down the problem step by step and solve it.

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🔹 Problem Summary:

We are given a normal distribution with:

- Mean $ \mu = 10 $
- Standard deviation $ \sigma = 2 $

And we're asked to find:

1. The z-score for $ x = 13 $
2. The area (probability) to the left of $ x = 13 $
3. The area to the right of $ x = 13 $
4. The percentile corresponding to $ x = 13 $

Also, there’s a shaded region under the curve from $ x = 10 $ to $ x = 13 $, and we’re told that the total area under the curve is 1.

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🔸 Step 1: Find the z-score

The z-score formula is:

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

Plug in the values:

$$
z = \frac{13 - 10}{2} = \frac{3}{2} = 1.5
$$

✔ So, the z-score is 1.5

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🔸 Step 2: Area to the left of $ x = 13 $ (i.e., $ P(X < 13) $)

We need to find the cumulative probability for $ z = 1.5 $

From the standard normal distribution table:

$$
P(Z < 1.5) = 0.9332
$$

So, about 93.32% of the data lies below $ x = 13 $

✔ Area to the left = 0.9332

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🔸 Step 3: Area to the right of $ x = 13 $

This is:

$$
P(X > 13) = 1 - P(X < 13) = 1 - 0.9332 = 0.0668
$$

✔ Area to the right = 0.0668

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🔸 Step 4: Percentile of $ x = 13 $

The percentile is the percentage of data below $ x $. Since $ P(X < 13) = 0.9332 $, this corresponds to the 93.32nd percentile.

✔ Percentile = 93.32th

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🔸 Step 5: Area between $ x = 10 $ and $ x = 13 $

We want $ P(10 < X < 13) $

Convert both to z-scores:

- $ x = 10 $ → $ z = \frac{10 - 10}{2} = 0 $
- $ x = 13 $ → $ z = 1.5 $

Now find:

$$
P(0 < Z < 1.5) = P(Z < 1.5) - P(Z < 0) = 0.9332 - 0.5 = 0.4332
$$

✔ So, area between 10 and 13 is 0.4332 or 43.32%

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✔ Final Answers:

| Question | Answer |
|--------|--------|
| 1. z-score for $ x = 13 $ | $ z = 1.5 $ |
| 2. Area to the left of $ x = 13 $ | $ 0.9332 $ |
| 3. Area to the right of $ x = 13 $ | $ 0.0668 $ |
| 4. Percentile of $ x = 13 $ | $ 93.32^{\text{nd}} $ percentile |
| 5. Area between $ x = 10 $ and $ x = 13 $ | $ 0.4332 $ |

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📌 Explanation of Shaded Region:

In the graph, the shaded region from $ x = 10 $ to $ x = 13 $ represents the probability that a randomly selected value from this normal distribution falls between the mean and 13. This is half of the distribution (from mean) plus the tail up to 1.5 standard deviations above the mean — hence, 43.32%.

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Let me know if you'd like a visual explanation or help with similar problems!