Compute the range and sample standard deviation for strength of the concrete​ (in psi). 3910​, 4120​, 3100​, 3000​, 2980​, 3820​, 4120​, 4060 Question content area bottom Part 1 The range is 1140 psi. Part 2 sequals=enter your response here psi​ (Round to one decimal place as​ needed.)


To solve this problem, we'll first verify the given range and then calculate the sample standard deviation for the provided concrete strength data.

### 1. Calculate the Range

The range is calculated as the difference between the maximum and minimum values in the data set.

Given data (in psi): \(3910, 4120, 3100, 3000, 2980, 3820, 4120, 4060\).

- Maximum value: \(4120\) psi
- Minimum value: \(2980\) psi

\[
\text{Range} = \text{Maximum} - \text{Minimum} = 4120 - 2980 = 1140 \text{ psi}
\]

The given range of 1140 psi is correct.

### 2. Calculate the Sample Standard Deviation

To calculate the sample standard deviation, use the following steps:

1. **Find the mean (average) of the data.**

\[
\text{Mean} = \frac{\sum x_i}{n}
\]

Where \( x_i \) are the data points, and \( n \) is the number of data points.

\[
\text{Mean} = \frac{3910 + 4120 + 3100 + 3000 + 2980 + 3820 + 4120 + 4060}{8}
\]

\[
\text{Mean} = \frac{29110}{8} = 3638.75 \text{ psi}
\]

2. **Find the squared differences from the mean.**

\[
(3910 - 3638.75)^2 = 73714.06
\]
\[
(4120 - 3638.75)^2 = 23152.56
\]
\[
(3100 - 3638.75)^2 = 290775.06
\]
\[
(3000 - 3638.75)^2 = 407651.56
\]
\[
(2980 - 3638.75)^2 = 433400.06
\]
\[
(3820 - 3638.75)^2 = 32760.06
\]
\[
(4120 - 3638.75)^2 = 23152.56
\]
\[
(4060 - 3638.75)^2 = 17722.56
\]

3. **Sum of the squared differences.**

\[
73714.06 + 23152.56 + 290775.06 + 407651.56 + 433400.06 + 32760.06 + 23152.56 + 17722.56 = 1307328.4
\]

4. **Divide by \(n - 1\) (since it's a sample standard deviation).**

\[
\text{Variance} = \frac{1307328.4}{8-1} = \frac{1307328.4}{7} = 186761.2
\]

5. **Take the square root to find the standard deviation.**

\[
\text{Standard Deviation} = \sqrt{186761.2} \approx 432.2 \text{ psi}
\]

### Answer
- **Range**: \( 1140 \) psi
- **Sample Standard Deviation**: \( 432.2 \) psi (rounded to one decimal place)

Would you like more details or have any questions?

### Further Questions
1. How is sample standard deviation different from population standard deviation?
2. What are some practical implications of having a high standard deviation in concrete strength?
3. How would the range and standard deviation change if one data point were significantly higher?
4. Can you provide a visual representation of this data using a histogram or box plot?
5. How would you calculate the variance from the standard deviation?

**Tip:** Always use the sample standard deviation when dealing with a subset of data to estimate the variability of a larger population.

Learn how to calculate the range and sample standard deviation for concrete strength data, including a step-by-step solution and detailed calculations. Explore the variability in concrete strength based on given psi values. Suitable for higher education and professionals.

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