To compute the range and the sample standard deviation for the strength of the concrete given by the values [tex]\( 3910, 4140, 3500, 3200, 2950, 3850, 4140, 4020 \)[/tex]:
#### Range Calculation:
1. Identify the maximum value: From the given values, the maximum value is 4140.
2. Identify the minimum value: From the given values, the minimum value is 2950.
3. Calculate the range:
[tex]\[
\text{Range} = \text{Maximum Value} - \text{Minimum Value} = 4140 - 2950 = 1190
\][/tex]
So, the range of the concrete strengths is 1190 psi.
#### Sample Standard Deviation Calculation:
1. List the given values: [tex]\( 3910, 4140, 3500, 3200, 2950, 3850, 4140, 4020 \)[/tex].
2. Calculate the mean (average) of the values:
[tex]\[
\bar{x} = \frac{3910 + 4140 + 3500 + 3200 + 2950 + 3850 + 4140 + 4020}{8} = \frac{29710}{8} = 3713.75
\][/tex]
3. Calculate each value's deviation from the mean and square it:
[tex]\[
\begin{align*}
(3910 - 3713.75)^2 &= (196.25)^2 = 38513.0625 \\
(4140 - 3713.75)^2 &= (426.25)^2 = 181690.0625 \\
(3500 - 3713.75)^2 &= (-213.75)^2 = 45690.0625 \\
(3200 - 3713.75)^2 &= (-513.75)^2 = 263950.5625 \\
(2950 - 3713.75)^2 &= (-763.75)^2 = 583317.0625 \\
(3850 - 3713.75)^2 &= (136.25)^2 = 18559.0625 \\
(4140 - 3713.75)^2 &= (426.25)^2 = 181690.0625 \\
(4020 - 3713.75)^2 &= (306.25)^2 = 93790.0625 \\
\end{align*}
\][/tex]
4. Sum these squared deviations:
[tex]\[
38513.0625 + 181690.0625 + 45690.0625 + 263950.5625 + 583317.0625 + 18559.0625 + 181690.0625 + 93790.0625 = 1408200.5
\][/tex]
5. Calculate the sample variance:
[tex]\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{1408200.5}{7} = 201171.5
\][/tex]
6. Calculate the sample standard deviation:
[tex]\[
s = \sqrt{s^2} = \sqrt{201171.5} \approx 448.35 \approx 448.4 \text{ (rounded to one decimal place)}
\][/tex]
So, the sample standard deviation of the concrete strengths is approximately 448.4 psi.
#### Final Results:
- The range of the concrete strengths is [tex]\( 1190 \)[/tex] psi.
- The sample standard deviation of the concrete strengths is [tex]\( 448.4 \)[/tex] psi (rounded to one decimal place).
