Solved STANDARD NORMAL DISTRIBUTION WORKSHEET NAME: In each ... - Free Printable

You're working on a Standard Normal Distribution Worksheet, which involves finding probabilities (or z-scores) using the standard normal distribution (mean = 0, standard deviation = 1). You need to:

- Shade the appropriate region under the curve (left or right of a z-value).
- Find the value of `a`, which is either:
- A probability (for parts a–e), or
- A z-score (for parts f–h).

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Let’s solve each part step by step.

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a) P(z < -1.2) = a

This is the area to the left of z = -1.2.

✔ Shade: Left tail, from -∞ to -1.2.

Use a Z-table or calculator:

> From standard normal table:
> P(z < -1.2) = 0.1151

✔ Answer: a = 0.1151

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b) P(z > .01) = a

This is the area to the right of z = 0.01.

✔ Shade: Right tail, from 0.01 to ∞.

We use:
P(z > 0.01) = 1 - P(z < 0.01)

From Z-table:
P(z < 0.01) = 0.5040
→ So, P(z > 0.01) = 1 - 0.5040 = 0.4960

✔ Answer: a = 0.4960

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c) P(z < .45) = a

Area to the left of z = 0.45.

✔ Shade: Left side up to 0.45.

From Z-table:
P(z < 0.45) = 0.6736

✔ Answer: a = 0.6736

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d) P(z < .33) = a

Area to the left of z = 0.33.

✔ Shade: Left side up to 0.33.

From Z-table:
P(z < 0.33) = 0.6293

✔ Answer: a = 0.6293

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e) P(z > 2.33) = a

Area to the right of z = 2.33.

✔ Shade: Right tail from 2.33 to ∞.

P(z > 2.33) = 1 - P(z < 2.33)

From Z-table:
P(z < 2.33) = 0.9901
→ So, P(z > 2.33) = 1 - 0.9901 = 0.0099

✔ Answer: a = 0.0099

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f) P(z > a) = .6660

Here, you’re given the probability and must find the z-score `a`.

Since P(z > a) = 0.6660, that means more than half the area is to the right → so `a` must be negative (because for positive z, P(z > a) < 0.5).

We can rewrite:

P(z > a) = 0.6660
→ P(z < a) = 1 - 0.6660 = 0.3340

Now find the z-score such that P(z < a) = 0.3340.

Look up 0.3340 in the Z-table.

Closest value:
Z = -0.43 → P(z < -0.43) ≈ 0.3336
Z = -0.42 → P(z < -0.42) ≈ 0.3372

Interpolate or choose closest:

0.3336 is very close to 0.3340 → use a ≈ -0.43

✔ Answer: a ≈ -0.43

*(Note: Some tables may give slightly different values — if your table shows 0.3340 at z = -0.428, you could use that, but -0.43 is fine for most purposes.)*

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g) P(z < a) = .3336

Given probability, find z-score `a`.

Look up 0.3336 in the Z-table.

From standard table:
P(z < -0.43) = 0.3336

✔ Answer: a = -0.43

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h) P(z < a) = .8289

Find z-score where cumulative probability is 0.8289.

Look up 0.8289 in Z-table.

Check:

- z = 0.95 → P(z < 0.95) = 0.8289 ✔ Exact match!

✔ Answer: a = 0.95

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## ✔ Final Answers Summary:

| Problem | Answer |
|---------|--------------|
| a) | 0.1151 |
| b) | 0.4960 |
| c) | 0.6736 |
| d) | 0.6293 |
| e) | 0.0099 |
| f) | -0.43 |
| g) | -0.43 |
| h) | 0.95 |

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## 📌 Shading Instructions Recap:

- Left of z: Shade from left end to the z-value.
- Right of z: Shade from z-value to right end.
- For a), c), d), g), h) → shade left.
- For b), e), f) → shade right.

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Let me know if you want help drawing the shaded curves or using a specific Z-table!