Questions: 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. Range = day(s) s= (Round to two decimal places as needed.)

Solution Steps

To determine the range, subtract the smallest value from the largest value in the dataset. For the sample standard deviation, calculate the mean of the dataset, find the squared differences from the mean for each data point, sum these squared differences, divide by the number of data points minus one, and then take the square root of the result.

The range \( R \) is calculated as follows: \[ R = \max(\text{times to hatch}) - \min(\text{times to hatch}) = 16 - 10 = 6 \]

The sample standard deviation \( s \) is calculated using the formula: \[ s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} \] where \( \bar{x} \) is the sample mean. First, we find the mean: \[ \bar{x} = \frac{12 + 15 + 10 + 14 + 13 + 11 + 16}{7} = \frac{91}{7} = 13 \] Next, we calculate the squared differences: \[ \begin{align_} (12 - 13)^2 & = 1 \\ (15 - 13)^2 & = 4 \\ (10 - 13)^2 & = 9 \\ (14 - 13)^2 & = 1 \\ (13 - 13)^2 & = 0 \\ (11 - 13)^2 & = 4 \\ (16 - 13)^2 & = 9 \\ \end{align_} \] Summing these squared differences: \[ \sum (x_i - \bar{x})^2 = 1 + 4 + 9 + 1 + 0 + 4 + 9 = 28 \] Now, substituting into the formula for \( s \): \[ s = \sqrt{\frac{28}{7-1}} = \sqrt{\frac{28}{6}} = \sqrt{\frac{14}{3}} \approx 2.1602 \]

Final Answer