Question

a study recorded the time it took for a sample of seven different species of frogs and toads eggs to hatch. the following table shows the times to hatch, in days. determine the range and sample standard deviation. 14 13 6 6 14 13 5 range = day(s) s = (round to two decimal places as needed.)

Explanation:

Step1: Find the maximum and minimum values

The data set is \(14, 13, 6, 6, 14, 13, 5\). The maximum value \(x_{max}=14\) and the minimum value \(x_{min}=5\).

Step2: Calculate the range

The range \(R\) is given by the formula \(R = x_{max}-x_{min}\). So \(R=14 - 5=9\).

Step3: Calculate the mean \(\bar{x}\)

\(\bar{x}=\frac{14 + 13+6+6+14+13+5}{7}=\frac{71}{7}\approx10.14\).

Step4: Calculate the squared - differences \((x_i-\bar{x})^2\)

For \(x_1 = 14\): \((14 - 10.14)^2=(3.86)^2 = 14.8996\)
For \(x_2 = 13\): \((13 - 10.14)^2=(2.86)^2 = 8.1796\)
For \(x_3 = 6\): \((6 - 10.14)^2=(- 4.14)^2 = 17.1396\)
For \(x_4 = 6\): \((6 - 10.14)^2=(-4.14)^2 = 17.1396\)
For \(x_5 = 14\): \((14 - 10.14)^2=(3.86)^2 = 14.8996\)
For \(x_6 = 13\): \((13 - 10.14)^2=(2.86)^2 = 8.1796\)
For \(x_7 = 5\): \((5 - 10.14)^2=(-5.14)^2 = 26.4196\)
The sum of squared - differences \(\sum_{i = 1}^{7}(x_i-\bar{x})^2=14.8996+8.1796 + 17.1396+17.1396+14.8996+8.1796+26.4196=106.8572\).

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

The formula for the sample standard deviation is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\), where \(n = 7\).
\(s=\sqrt{\frac{106.8572}{7 - 1}}=\sqrt{\frac{106.8572}{6}}\approx\sqrt{17.8095}\approx4.22\).

Answer:

Range = 9 day(s)
s = 4.22