Normal Distributions Worksheet 3 | PDF | Standard Deviation ... - Free Printable

Problem Analysis:

The worksheet involves working with normal distributions. Here's a breakdown of the tasks:

1. Finding values based on distances from the mean:
- Given a normal distribution with a mean (\(\mu\)) of 45 and a standard deviation (\(\sigma\)) of 8.3, we need to calculate values that are specific distances (in terms of standard deviations) from the mean.

2. Sketching normal curves:
- For each given mean and standard deviation, we need to sketch a normal curve and label the \(x\)-axis at one, two, and three standard deviations from the mean.

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Solution:

#### Part 1: Finding Values Based on Distances from the Mean

Given:
- Mean (\(\mu\)) = 45
- Standard deviation (\(\sigma\)) = 8.3

We use the formula:
\[ \text{Value} = \mu + k \cdot \sigma \]
where \(k\) is the number of standard deviations from the mean.

1. +1 standard deviation from the mean:
\[
\text{Value} = \mu + 1 \cdot \sigma = 45 + 1 \cdot 8.3 = 45 + 8.3 = 53.3
\]

2. +3 standard deviations from the mean:
\[
\text{Value} = \mu + 3 \cdot \sigma = 45 + 3 \cdot 8.3 = 45 + 24.9 = 69.9
\]

3. -1 standard deviation from the mean:
\[
\text{Value} = \mu - 1 \cdot \sigma = 45 - 1 \cdot 8.3 = 45 - 8.3 = 36.7
\]

4. -2 standard deviations from the mean:
\[
\text{Value} = \mu - 2 \cdot \sigma = 45 - 2 \cdot 8.3 = 45 - 16.6 = 28.4
\]

So, the answers are:
1. \(53.3\)
2. \(69.9\)
3. \(36.7\)
4. \(28.4\)

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#### Part 2: Sketching Normal Curves

For each normal distribution, we sketch a bell-shaped curve and label the \(x\)-axis at one, two, and three standard deviations from the mean. The general steps are:

1. Draw a symmetric bell-shaped curve centered at the mean.
2. Mark the mean on the \(x\)-axis.
3. Label points on the \(x\)-axis at:
- One standard deviation from the mean: \(\mu \pm \sigma\)
- Two standard deviations from the mean: \(\mu \pm 2\sigma\)
- Three standard deviations from the mean: \(\mu \pm 3\sigma\)

Let's go through each case:

5. Mean = 95, Standard Deviation = 12:
- One standard deviation: \(95 \pm 12\) → \(83, 107\)
- Two standard deviations: \(95 \pm 24\) → \(71, 119\)
- Three standard deviations: \(95 \pm 36\) → \(59, 131\)

6. Mean = 100, Standard Deviation = 15:
- One standard deviation: \(100 \pm 15\) → \(85, 115\)
- Two standard deviations: \(100 \pm 30\) → \(70, 130\)
- Three standard deviations: \(100 \pm 45\) → \(55, 145\)

7. Mean = 60, Standard Deviation = 6:
- One standard deviation: \(60 \pm 6\) → \(54, 66\)
- Two standard deviations: \(60 \pm 12\) → \(48, 72\)
- Three standard deviations: \(60 \pm 18\) → \(42, 78\)

8. Mean = 23.8, Standard Deviation = 5.2:
- One standard deviation: \(23.8 \pm 5.2\) → \(18.6, 29.0\)
- Two standard deviations: \(23.8 \pm 10.4\) → \(13.4, 34.2\)
- Three standard deviations: \(23.8 \pm 15.6\) → \(8.2, 40.0\)

9. Mean = 676, Standard Deviation = 60:
- One standard deviation: \(676 \pm 60\) → \(616, 736\)
- Two standard deviations: \(676 \pm 120\) → \(556, 796\)
- Three standard deviations: \(676 \pm 180\) → \(496, 856\)

10. Mean = 54.2, Standard Deviation = 12.3:
- One standard deviation: \(54.2 \pm 12.3\) → \(41.9, 66.5\)
- Two standard deviations: \(54.2 \pm 24.6\) → \(29.6, 78.8\)
- Three standard deviations: \(54.2 \pm 36.9\) → \(17.3, 91.1\)

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Final Answers:

1. \(53.3\)
2. \(69.9\)
3. \(36.7\)
4. \(28.4\)

For the sketches, follow the steps above to label the \(x\)-axis for each normal curve.

\[
\boxed{53.3, 69.9, 36.7, 28.4}
\]