he 162 heights of males from a data set of body measurements vary from a low of 151.0cm to a high of 193.3193.3 cm. Use the range rule of thumb to estimate the standard deviation s and compare the result to the standard deviation of 12.73 cm calculated using the 162 heights. What does the result suggest about the accuracy of estimates of s found using the range rule of​ thumb? Assume the estimate is accurate if it is within 1.11.1 cm.
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Part 1
Use the range rule of thumb to estimate the standard deviation s.
sequals=enter your response here cm
​(Type an integer or decimal rounded to two decimal places as​ needed.)

The **range rule of thumb** estimates the standard deviation (\(s\)) of a dataset by using the following formula:

\[
s \approx \frac{\text{Range}}{4}
\]

Where the **range** is the difference between the maximum and minimum values in the dataset.

### Step 1: Calculate the range

\[
\text{Range} = \text{Maximum height} - \text{Minimum height}
\]
\[
\text{Range} = 193.3 \, \text{cm} - 151.0 \, \text{cm} = 42.3 \, \text{cm}
\]

### Step 2: Apply the range rule of thumb

\[
s \approx \frac{42.3}{4} = 10.575 \, \text{cm}
\]

### Step 3: Compare the result

The calculated standard deviation using the actual data is \(12.73 \, \text{cm}\). The estimate using the range rule of thumb is \(10.58 \, \text{cm}\).

### Step 4: Accuracy check

To check if the estimate is accurate, see if the estimate is within \(1.1 \, \text{cm}\) of the actual standard deviation:

\[
\left|12.73 - 10.58\right| = 2.15 \, \text{cm}
\]

Since \(2.15 \, \text{cm}\) is greater than \(1.1 \, \text{cm}\), the range rule of thumb estimate is **not accurate** in this case.

### Conclusion:
The result suggests that the range rule of thumb may not always provide an accurate estimate for the standard deviation, especially when compared to the actual calculated value.

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Would you like more details or explanations on any part of this?

Here are 5 related questions:
1. How can the range rule of thumb be adjusted to improve accuracy?
2. What are the common limitations of using the range rule of thumb for standard deviation estimation?
3. Why does the range rule of thumb often underestimate the standard deviation in certain data sets?
4. What methods other than the range rule of thumb can be used to estimate standard deviation?
5. How would outliers affect the accuracy of the range rule of thumb?

**Tip:** The range rule of thumb works best for symmetric, unimodal distributions, but it can be inaccurate for skewed or multimodal data.

The 162 heights of males from a data set of body measurements vary from a low of 151.0cm to a high of 193.3 cm. Use the range rule of thumb to estimate the standard deviation s and compare the result to the standard deviation of 12.73 cm calculated using the 162 heights. What does the result suggest about the accuracy of estimates of s found using the range rule of thumb? Assume the estimate is accurate if it is within 1.1 cm.

Explore how to estimate the standard deviation using the range rule of thumb with data from male heights ranging from 151.0 cm to 193.3 cm. Compare this estimate to the actual calculated standard deviation of 12.73 cm and analyze the accuracy of the rule. Suitable for students in Grades 10-12.

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