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.
7 10 13 10 13 7 5
range = 8 day(s)
s = □ (round to two decimal places as needed.)

Explanation:

Step1: Calculate the mean

The data set is \(7, 10, 13, 10, 13, 7, 5\). The mean \(\bar{x}=\frac{7 + 10+13+10+13+7+5}{7}=\frac{65}{7}\approx9.29\).

Step2: Calculate the squared - differences

For \(x_1 = 7\): \((7 - 9.29)^2=(- 2.29)^2 = 5.2441\)
For \(x_2 = 10\): \((10 - 9.29)^2=(0.71)^2 = 0.5041\)
For \(x_3 = 13\): \((13 - 9.29)^2=(3.71)^2 = 13.7641\)
For \(x_4 = 10\): \((10 - 9.29)^2=(0.71)^2 = 0.5041\)
For \(x_5 = 13\): \((13 - 9.29)^2=(3.71)^2 = 13.7641\)
For \(x_6 = 7\): \((7 - 9.29)^2=(-2.29)^2 = 5.2441\)
For \(x_7 = 5\): \((5 - 9.29)^2=(-4.29)^2 = 18.4041\)

Step3: Calculate the sum of squared - differences

\(\sum_{i = 1}^{7}(x_i-\bar{x})^2=5.2441+0.5041 + 13.7641+0.5041+13.7641+5.2441+18.4041=57.4327\)

Step4: Calculate the sample variance

The sample variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\), where \(n = 7\). So \(s^2=\frac{57.4327}{7 - 1}=\frac{57.4327}{6}\approx9.5721\)

Step5: Calculate the sample standard deviation

The sample standard deviation \(s=\sqrt{s^2}=\sqrt{9.5721}\approx3.10\)

Answer:

3.10