Question

for the sample data set below, determine the
a. range.
b. sample standard deviation.

2, 3, 9, 1, 7

a. the range is \boxed{}.
(simplify your answer.)

b. the sample standard deviation is \boxed{}.
(round to one decimal place as needed.)

Explanation:

Step1: Find the range

The range is calculated as the difference between the maximum and minimum values in the data set.
The data set is \(2,3,9,1,7\). The maximum value \(x_{max}=9\) and the minimum value \(x_{min} = 1\).

$$ \text{Range}=x_{max}-x_{min}=9 - 1 $$

Step2: Calculate the sample mean \(\bar{x}\)

The formula for the sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 5\) (the number of data points), and \(x_{i}\) are the data values.

$$ \bar{x}=\frac{2 + 3+9+1+7}{5}=\frac{22}{5}=4.4 $$

Step3: Calculate the squared differences \((x_{i}-\bar{x})^{2}\)

For \(x_1 = 2\): \((2 - 4.4)^{2}=(- 2.4)^{2}=5.76\)
For \(x_2 = 3\): \((3 - 4.4)^{2}=(-1.4)^{2}=1.96\)
For \(x_3 = 9\): \((9 - 4.4)^{2}=(4.6)^{2}=21.16\)
For \(x_4 = 1\): \((1 - 4.4)^{2}=(-3.4)^{2}=11.56\)
For \(x_5 = 7\): \((7 - 4.4)^{2}=(2.6)^{2}=6.76\)

Step4: Calculate the sum of squared differences \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\)

$$ \sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=5.76+1.96 + 21.16+11.56+6.76=47.2 $$

Step5: Calculate the sample variance \(s^{2}\)

The formula for the sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), with \(n=5\)

$$ s^{2}=\frac{47.2}{5-1}=\frac{47.2}{4}=11.8 $$

Step6: Calculate the sample standard deviation \(s\)

The sample standard deviation \(s=\sqrt{s^{2}}\)

$$ s=\sqrt{11.8}\approx3.4 $$

Answer:

a. \(8\)
b. \(3.4\)