Questions: For the data shown, answer the questions. Round to 2 decimal places. x 15.2 13.4 7.8 7.4 11.1 7.3 7.1 13.5 27.9 10 Find the mean: Find the median: Find the sample standard deviation:

Solution Steps

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

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

Substituting the values:

\[ \mu = \frac{120.7}{10} = 12.07 \]

First, we sort the data:

\[ \text{Sorted data} = [7.1, 7.3, 7.4, 7.8, 10, 11.1, 13.4, 13.5, 15.2, 27.9] \]

To find the median \( Q \), we calculate the rank:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (10 + 1) = 5.5 \]

Since the rank is not an integer, we take the average of the values at positions 5 and 6:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{10 + 11.1}{2} = 10.55 \]

First, we calculate the variance \( \sigma^2 \) using the formula:

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

Substituting the values, we find:

\[ \sigma^2 = 39.66 \]

Then, the sample standard deviation \( \sigma \) is calculated as:

\[ \sigma = \sqrt{39.66} = 6.3 \]

Final Answer