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.