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 $x_i$ $(x_i - ?)^2$ 1 10 $(10 - )^2=0.36$ 2 10 0.36 3 15 4 7 12.96 5 = ? 0.16 average: ? = total = variance: ? = total/? = st. dev.: ? = question help: video written example message instructor post to forum submit question

Explanation:

Step1: Calculate the sample mean

The sample data is \(x_1 = 10,x_2 = 10,x_3 = 15,x_4 = 7,x_5 = 11\). The formula for the sample mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\), where \(n = 5\).
\(\bar{x}=\frac{10 + 10+15+7+11}{5}=\frac{53}{5}=10.6\)

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

• 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 \((x_i-\bar{x})^2\)

\(Total=0.36 + 0.36+19.36+12.96+0.16=33.2\)

Step4: 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 \(n = 5\), so \(s^2=\frac{33.2}{5 - 1}=\frac{33.2}{4}=8.3\)

Step5: Calculate the sample standard - deviation \(s\)

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

Answer:

\(2.88\)