Questions: Given below is a data set: 75 66 54 78 64 58 69 53 65 70 74 65 74 65 58 68 55 65 88 68 84 53 63 51 84 67 64 50 61 68 Examine the data above by reviewing the stem-and-leaf plot. Here is the start of the plot. missing values? 5# 01334588 6# 13445555678889 7# 04458 8#

Solution Steps

The mean \( \mu \) of the dataset is calculated using the formula:

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

For the given dataset, the sum of the values is \( 1977 \) and the number of values \( N = 30 \). Thus, the mean is:

\[ \mu = \frac{1977}{30} = 65.9 \]

To find the median, we first sort the data:

\[ \text{Sorted data} = [50, 51, 53, 53, 54, 55, 58, 58, 61, 63, 64, 64, 65, 65, 65, 65, 66, 67, 68, 68, 68, 69, 70, 74, 74, 75, 78, 84, 84, 88] \]

Since \( N = 30 \) (even), the median \( Q \) is calculated as:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (30 + 1) = 15.5 \]

This means we take the average of the 15th and 16th values:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{65 + 65}{2} = 65.0 \]

The variance \( \sigma^2 \) is calculated using the formula:

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

Substituting the values, we find:

\[ \sigma^2 = 95.89 \]

The standard deviation \( \sigma \) is then:

\[ \sigma = \sqrt{95.89} \approx 9.79 \]

Final Answer