https://file.aiquickdraw.com/mathbot/file/1727930233281d0svfagr.jpg

Let's go through the questions step-by-step, based on the image you provided:

### 2(a) Fill in the blanks of the empirical rule diagram:
From the image, we have a normal distribution with the mean (μ) of 5.5 feet, and the standard deviation (σ) of 3.5 inches (0.2917 feet).

Using the empirical rule (68-95-99.7 rule), which tells us about the distribution of dataLet's go through the questions step-by-step, based on the image you provided:

### 2(a) Fill in the blanks of the empirical rule diagram:
From the image, we have a normal distribution with the mean (μ) of 5.5 feet, and the standard deviation (σ) of 3.5 inches (0.2917 feet).

Using the empirical rule (68-95-99.7 rule), which tells us about the distribution of 
- 68% of data falls within 1 standard deviation of the mean
- 95% falls within 2 standard deviations
- 99.7% falls within 3 standard deviations.

Here are the values in the diagram based on this rule:
- \( \mu - 3\sigma = 5.5 - (3 \times 0.2917) = 5 - 0.8751 \approx 5.0 \) feet
- \( \mu - 2\sigma = 5.5 - (2 \times 0.2917) = 5.5 - 0.5834 \approx 5.1 \) feet
- \( \mu - \sigma = 5.5 - 0.2917 \approx 5.2 \) feet
- \( \mu = 5.5 \) feet
- \( \mu + \sigma = 5.5 + 0.2917 \approx 5.8 \) feet
- \( \mu + 2\sigma = 5.5 + 0.5834 \approx 6.1 \) feet
- \( \mu + 3\sigma = 5.5 + 0.8751 \approx 6.4 \) feet

So, the empirical rule for height is:

- \( 5.0 \text{ feet ( } \mu - 3\sigma \text{)} \)
- \( 5.1 \text{ feet ( } \mu - 2\sigma \text{)} \)
- \( 5.2 \text{ feet ( } \mu - 1\sigma \text{)} \)
- \( 5.5 \text{ feet (mean, } \mu \text{)} \)
- \( 5.8 \text{ feet ( } \mu + 1\sigma \text{)} \)
- \( 6.1 \text{ feet ( } \mu + 2\sigma \text{)} \)
- \( 6.4 \text{ feet ( } \mu + 3\sigma \text{)} \)

### 2(b) Shade the regions of the graph:
We need to find the probability that the height of an American adult female is between 5 feet 1.5 inches (5.125 feet) and 5 feet 8.5 inches (5.708 feet).

1. Convert 5 feet 1.5 inches to feet:  
   5 feet 1.5 inches = \( 5 + \frac{1.5}{12} = 5.125 \) feet.

2. Convert 5 feet 8.5 inches to feet:  
   5 feet 8.5 inches = \( 5 + \frac{8.5}{12} = 5.708 \) feet.

To find this probability, we need to look at the region between \( 5.125 \) feet and \( 5.708 \) feet. Both values lie between \( \mu - \sigma \) (which is 5.2 feet) and \( \mu + \sigma \) (which is 5.8 feet), encompassing the central 68% of the data. The height range is approximately within one standard deviation of the mean.

### 2(c) Find that probability:
Using the empirical rule:
- From \( \mu - \sigma \) to \( \mu + \sigma \) encompasses 68% of the data.
- Since we want the probability between 5.125 feet and 5.708 feet, this range approximately corresponds to about one standard deviation, so the probability is approximately **68%**.

### 3. Write your answer to Question 2c in \( P(\text{inequality}) = \) form:
The inequality corresponding to the height range is:

\[
P(5.125 \leq X \leq 5.708) \approx 0.68
\]

Let me know if you'd like more detailed steps, and feel free to ask questions!

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**Relative Questions:**
1. What is the empirical rule and how is it used in normal distribution problems?
2. How do we convert between feet and inches?
3. How do we interpret the standard deviation in a normal distribution curve?
4. What does the area under a normal distribution curve represent?
5. How can we calculate probabilities for other ranges in a normal distribution?

**Tip:** The empirical rule (68-95-99.7) is a powerful tool for estimating probabilities quickly when dealing with normal distributions!

Find the probability that a randomly chosen American adult female is between 5 feet 1.5 inches and 5 feet 8.5 inches, given a normal distribution with a mean height of 5 feet 5 inches and a standard deviation of 3.5 inches. Also, fill in the empirical rule diagram and write the probability in inequality form.

Learn how to calculate the probability that a randomly chosen American adult female is between 5 feet 1.5 inches and 5 feet 8.5 inches using the empirical rule and normal distribution. Understand the 68-95-99.7 rule and solve the problem with step-by-step solutions for students in Grades 9-12.

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