A certain type of concrete mix is designed to withstand 3000 pounds per square inch​ (psi) of pressure. The data below represent the strength of eight randomly selected
casts made out of the concrete. Compute the​ range, sample​ variance, and sample standard deviation for strength of the concrete​ (in psi).
3910​, 4060​, 3300​, 3000​, 2910​, 3840​, 4060​, 4040   
Question content area bottom
Part 1
The range is 1150 psi.
Part 2
s squareds2equals=enter your response here ​(Round to the nearest integer as​ needed.)

Let's calculate the sample variance and sample standard deviation step by step based on the data provided:

Concrete strengths (in psi):  
\[3910, 4060, 3300, 3000, 2910, 3840, 4060, 4040\]

### 1. Compute the sample mean (\(\bar{x}\)):

\[
\bar{x} = \frac{\sum x_i}{n}
\]

Where \(n = 8\) (the number of data points).

\[
\bar{x} = \frac{3910 + 4060 + 3300 + 3000 + 2910 + 3840 + 4060 + 4040}{8} = \frac{29120}{8} = 3640
\]

### 2. Compute the squared deviations from the mean:

For each data point, subtract the mean and square the result:

\[
(x_i - \bar{x})^2
\]

\[
(3910 - 3640)^2 = 72900, \quad (4060 - 3640)^2 = 176400, \quad (3300 - 3640)^2 = 115600
\]
\[
(3000 - 3640)^2 = 409600, \quad (2910 - 3640)^2 = 532900, \quad (3840 - 3640)^2 = 40000
\]
\[
(4060 - 3640)^2 = 176400, \quad (4040 - 3640)^2 = 160000
\]

### 3. Compute the sample variance (\(s^2\)):

The formula for sample variance is:

\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}
\]

Summing the squared deviations:

\[
\sum (x_i - \bar{x})^2 = 72900 + 176400 + 115600 + 409600 + 532900 + 40000 + 176400 + 160000 = 1685800
\]

Now divide by \(n - 1 = 8 - 1 = 7\):

\[
s^2 = \frac{1685800}{7} = 240829 \quad \text{(rounded to the nearest integer)}
\]

### 4. Compute the sample standard deviation (\(s\)):

The sample standard deviation is the square root of the sample variance:

\[
s = \sqrt{s^2} = \sqrt{240829} \approx 490.74
\]

### Final answers:

- **Sample variance** (\(s^2\)): 240829 psi\(^2\)
- **Sample standard deviation** (\(s\)): 490.74 psi (approximately)

Would you like further clarification or details on any part of the solution?

Here are some related questions you might find useful:
1. How is the sample variance different from population variance?
2. How does the range help in understanding the spread of the data?
3. What happens to the standard deviation if we increase the number of data points?
4. Why do we divide by \(n - 1\) in the formula for sample variance?
5. How can the standard deviation be interpreted in terms of data spread?

**Tip:** In statistics, a larger standard deviation indicates that the data points are more spread out from the mean.

A certain type of concrete mix is designed to withstand 3000 pounds per square inch​ (psi) of pressure. The data below represent the strength of eight randomly selected casts made out of the concrete. Compute the​ range, sample​ variance, and sample standard deviation for strength of the concrete​ (in psi). 3910​, 4060​, 3300​, 3000​, 2910​, 3840​, 4060​, 4040.

Learn how to compute the range, sample variance, and standard deviation for concrete strength data, including values like 3910, 4060, and 3300 psi. This problem covers key statistical concepts for high school students in Grades 10-12.

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