Question

the tourist bureau of the caribbean islands surveyed a sample of 6 united states tourists as they left to return home. the tourists were asked how many days they spent on their visits. their responses were as follows. 11, 4, 8, 10, 7, 8 find the standard deviation of this sample of numbers. round your answer to two decimal places. (if necessary, consult a list of formulas.)

Explanation:

Step1: Calculate the mean

The data set is \(11, 4, 8, 10, 7, 8\). The number of data points \(n = 6\).
The mean \(\bar{x}=\frac{11 + 4+8 + 10+7 + 8}{6}=\frac{48}{6} = 8\).

Step2: Calculate the squared differences from the mean

For each data point \(x_i\), calculate \((x_i-\bar{x})^2\):

• For \(x = 11\): \((11 - 8)^2=3^2 = 9\)
• For \(x = 4\): \((4 - 8)^2=(- 4)^2 = 16\)
• For \(x = 8\): \((8 - 8)^2=0^2 = 0\)
• For \(x = 10\): \((10 - 8)^2=2^2 = 4\)
• For \(x = 7\): \((7 - 8)^2=(-1)^2 = 1\)
• For \(x = 8\): \((8 - 8)^2=0^2 = 0\)

Step3: Calculate the sum of squared differences

Sum of \((x_i-\bar{x})^2=9 + 16+0 + 4+1 + 0=30\).

Step4: Calculate the sample variance

Sample variance \(s^2=\frac{\sum(x_i - \bar{x})^2}{n - 1}=\frac{30}{6-1}=\frac{30}{5}=6\).

Step5: Calculate the sample standard deviation

Sample standard deviation \(s=\sqrt{s^2}=\sqrt{6}\approx2.45\).

Answer:

\(2.45\)