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The question provides details on a normal distribution with a mean (\(\mu\)) of 73 and a standard deviation (\(\sigma\)) of 9. It asks for probabilities using the Empirical Rule, which is also known as the 68-95-99.7 rule. This rule states that:

1. 68% of the data falls within 1 standard deviation from the mean (i.e., between \(\mu - \sigma\) and \(\mu + \sigma\)).
2. 95% of the data falls within 2 standard deviations from the mean (i.e., between \(\mu - 2\sigma\) and \(\mu + 2\sigma\)).
3. 99.7% of the data falls within 3 standard deviations from the mean (i.e., between \(\mu - 3\sigma\) and \(\mu + 3\sigma\)).

### Let's calculate the values needed for the table:

#### 1. "Less than 55"
55 is \(\mu - 2\sigma\), where:
\[
\mu - 2\sigma = 73 - 2(9) = 73 - 18 = 55
\]
Based on the Empirical Rule, 95% of the data falls within two standard deviations. This leaves 5% of the data outside this range, but because the normal distribution is symmetric, half of this 5% (i.e., 2.5%) is below 55.

**Answer for less than 55:** 2.5%

#### 2. "Between 55 and 82"
82 is \(\mu + 1\sigma\), where:
\[
\mu + 1\sigma = 73 + 9 = 82
\]
According to the Empirical Rule, 68% of the data lies within one standard deviation (between 64 and 82). Since 55 is at \(\mu - 2\sigma\) and 82 is at \(\mu + 1\sigma\), we need to subtract the probability of being less than 55 (2.5%) from the total probability up to 82.

**Answer for between 55 and 82:**
\[
95\% - 2.5\% = 92.5\%
\]

#### 3. "Above 82"
82 is \(\mu + \sigma\), and since 68% of the data lies within one standard deviation of the mean, 32% lies outside this range. Symmetry of the distribution tells us that half of this 32% is above 82.

**Answer for above 82:** 16%

### Final table of probabilities:
- **Less than 55:** 2.5%
- **Between 55 and 82:** 92.5%
- **Above 82:** 16%

Do you want further details or explanations? 

Here are 5 related questions to deepen your understanding:
1. How does the Empirical Rule differ from using a standard normal distribution table?
2. What is the probability of scoring between 64 and 82?
3. How would you calculate probabilities for a non-symmetric distribution?
4. Can you find the z-score for a student scoring 85 on the test?
5. How would the probabilities change if the mean shifted to 80 but the standard deviation remained 9?

**Tip:** Always ensure that the data you are working with follows a normal distribution before applying the Empirical Rule.

A city school district has students take a mathematics test at the end of eighth grade to determine placement in high school math courses. The scores of this test are approximately normally distributed with a mean of 73 and a standard deviation of 9. Using the Empirical Rule, complete the table to show the probability, as a percent, that a random student receives a score less than 55, between 55 and 82, and above 82.

Solve a normal distribution problem involving a mathematics test with a mean of 73 and a standard deviation of 9. This problem requires using the Empirical Rule to calculate probabilities for scores less than 55, between 55 and 82, and above 82. Suitable for Grades 8-10.

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