Questions: Roller Coaster Heights The data show the heights in feet of roller coasters. Find the mean, median, midrange, and mode for the data. 152, 128, 50, 127, 73, 58, 131, 127, 56, 58, 95, 50, 152, 130 Part 1 of 4 (a) Find the mean. Rounding rule for the mean: round to one more decimal place than data, as needed. Mean:

Solution Steps

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

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

For the given roller coaster heights, the sum of the data is \(\sum_{i=1}^{14} x_i = 1387\) and the number of data points \(N = 14\). Thus, the mean is:

\[ \mu = \frac{1387}{14} = 99.1 \]

The median is the middle value of a dataset when it is ordered. If the dataset has an even number of observations, the median is the average of the two middle numbers. First, we sort the data:

Sorted data: \([50, 50, 56, 58, 58, 73, 95, 127, 127, 128, 130, 131, 152, 152]\)

The formula for the rank of the median in a dataset is:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (14 + 1) = 7.5 \]

Since the rank is not an integer, we use interpolation between the 7th and 8th values in the sorted list:

\[ Q = X_{\text{lower}} + w \times (X_{\text{upper}} - X_{\text{lower}}) \]

Where \(X_{\text{lower}} = 95\), \(X_{\text{upper}} = 127\), and \(w = 0.5\):

\[ Q = 95 + 0.5 \times (127 - 95) = 111.0 \]

The midrange is the average of the maximum and minimum values in the dataset:

\[ \text{Midrange} = \frac{\text{max} + \text{min}}{2} \]

For the given data, \(\text{max} = 152\) and \(\text{min} = 50\):

\[ \text{Midrange} = \frac{152 + 50}{2} = 101.0 \]

Final Answer