Given distribution of a score with a mean of 12 and standard deviation of 4. Assuming normality, (1) what percentage of cases fall between 8 and 16. (2) what percentage of cases lie above the score 18. (3) what percentage of scores lie below the score 6

To solve these problems, we'll use the properties of the normal distribution, which is defined by its mean (μ) and standard deviation (σ). Given:
- Mean (μ) = 12
- Standard Deviation (σ) = 4

We'll calculate the Z-scores for the specific values and then use the standard normal distribution table (or a calculator) to find the corresponding probabilities.

### (1) Percentage of cases between 8 and 16:
To find the percentage of cases between 8 and 16, we'll calculate the Z-scores for both 8 and 16.

- Z-score for 8:  
  \[
  Z = \frac{X - \mu}{\sigma} = \frac{8 - 12}{4} = \frac{-4}{4} = -1
  \]

- Z-score for 16:  
  \[
  Z = \frac{X - \mu}{\sigma} = \frac{16 - 12}{4} = \frac{4}{4} = 1
  \]

Now, we look up the Z-scores in the standard normal distribution table:

- The probability corresponding to \( Z = -1 \) is approximately 0.1587.
- The probability corresponding to \( Z = 1 \) is approximately 0.8413.

The percentage of cases between 8 and 16 is the difference between these probabilities:
\[
0.8413 - 0.1587 = 0.6826
\]
So, **68.26%** of the cases fall between 8 and 16.

### (2) Percentage of cases above 18:
First, we calculate the Z-score for 18:

- Z-score for 18:  
  \[
  Z = \frac{X - \mu}{\sigma} = \frac{18 - 12}{4} = \frac{6}{4} = 1.5
  \]

The probability corresponding to \( Z = 1.5 \) is approximately 0.9332.

The percentage of cases above 18 is:
\[
1 - 0.9332 = 0.0668
\]
So, **6.68%** of the cases lie above the score of 18.

### (3) Percentage of scores below 6:
Finally, we calculate the Z-score for 6:

- Z-score for 6:  
  \[
  Z = \frac{X - \mu}{\sigma} = \frac{6 - 12}{4} = \frac{-6}{4} = -1.5
  \]

The probability corresponding to \( Z = -1.5 \) is approximately 0.0668.

So, **6.68%** of the scores lie below the score of 6.

### Summary:
1. **68.26%** of cases fall between 8 and 16.
2. **6.68%** of cases lie above the score 18.
3. **6.68%** of scores lie below the score 6.

Would you like more details on any of these calculations or have any other questions?

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Here are five related questions for further exploration:
1. What percentage of scores lie between 10 and 14?
2. How would the percentages change if the standard deviation were 5 instead of 4?
3. What score corresponds to the 90th percentile in this distribution?
4. What is the Z-score for a score of 20?
5. How do you interpret the Z-score in a real-world context?

**Tip:** Understanding how to read and interpret Z-scores is crucial for working with normal distributions, as they allow you to standardize any normal distribution to compare different datasets.

Explore how to calculate percentages in a normal distribution with a mean of 12 and standard deviation of 4. Learn to find probabilities between specific score ranges and above or below given scores using Z-scores and the standard normal distribution table.

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