Questions: Lengths of a random sample of 15 rivers on the South Island of New Zealand that flow to the Tasman Sea (km or kilometers) are listed in the table below. Length 40 76 56 72 68 177 64 56 32 64 80 35 32 56 80 For the data shown above, find the following. Round answer in the first blank to 1 decimal place(s). In the second blank put the correct units. Find the mean: Find the median: Find the range: Find the standard deviation:

Solution Steps

To find the mean \( \mu \) of the river lengths, we use the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \]

where \( N \) is the number of rivers and \( x_i \) are the lengths of the rivers. For our data:

\[ \mu = \frac{988}{15} = 65.9 \]

To find the median, we first sort the river lengths:

\[ \text{Sorted data: } [32, 32, 35, 40, 56, 56, 56, 64, 64, 68, 72, 76, 80, 80, 177] \]

The formula for the rank of the median is:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (15 + 1) = 8.0 \]

The quantile is at position 8, which corresponds to the value:

\[ \text{Median} = 64 \]

The range is calculated as the difference between the maximum and minimum river lengths:

\[ \text{Range} = \max(x) - \min(x) = 177 - 32 = 145 \]

To find the standard deviation, we first calculate the variance \( \sigma^2 \) using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \]

Substituting the values, we find:

\[ \sigma^2 = 1222.1 \]

Then, the standard deviation \( \sigma \) is:

\[ \sigma = \sqrt{1222.1} = 35.0 \]

Final Answer