Question

the numbers of false fire alarms were counted each month at a sample of sites. the results are given in the following table. 10 10 15 7 11 compute the value of the sample standard deviation using the table below: (round the answers to 2 decimal places) i $x_i$ $(x_i - \text{?})^2$ 1 10 $(10 - square)^2 = 0.36$ 2 10 0.36 3 15 $square$ 4 7 12.96 5 = $\text{?}$ $square$ 0.16 average: $\text{?}$ = $square$ total = $square$ variance: $\text{?}$ = total/$\text{?}$ = $square$ st. dev.: $\text{?}$ = $square$ question help: video written example message instructor post to forum submit question

Explanation:

Step1: Calculate the average

The data set is \(10, 10, 15, 7, 11\). The sum of the data is \(10 + 10+15 + 7+11=53\). There are \(n = 5\) data - points. The average \(\bar{x}=\frac{53}{5}=10.6\).

Step2: Calculate \((x_i-\bar{x})^2\) for each data - point

• For \(x_1 = 10\), \((10 - 10.6)^2=(- 0.6)^2 = 0.36\)
• For \(x_2 = 10\), \((10 - 10.6)^2=(-0.6)^2 = 0.36\)
• For \(x_3 = 15\), \((15 - 10.6)^2=(4.4)^2 = 19.36\)
• For \(x_4 = 7\), \((7 - 10.6)^2=(-3.6)^2 = 12.96\)
• For \(x_5 = 11\), \((11 - 10.6)^2=(0.4)^2 = 0.16\)

Step3: Calculate the total of \((x_i-\bar{x})^2\)

The total is \(0.36+0.36 + 19.36+12.96+0.16=33.2\)

Step4: Calculate the sample variance

The sample variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\). Here, \(n = 5\), so \(s^2=\frac{33.2}{4}=8.3\)

Step5: Calculate the sample standard deviation

The sample standard deviation \(s=\sqrt{s^2}=\sqrt{8.3}\approx2.88\)

Answer:

\(2.88\)