Questions: The data set represents wait times (in minutes) for various services at a state's Department of Motor Vehicles locations. Which wait time represents the 50th percentile? How would you interpret this? 27 4 8 4 3 26 18 21 1 3 3 5 5 6 10 1 22 23 10 6 7 2 1 6 6 2 4 14 15 16 4 19 3 19 26 5 3 4 7 6 10 9 10 20 18 3 20 10 Which wait time represents the 50th percentile? 6 minutes (Type a whole number.)

Solution Steps

The given dataset of wait times (in minutes) is as follows:

\[ \{27, 4, 8, 4, 3, 26, 18, 21, 1, 3, 3, 5, 5, 6, 10, 1, 22, 23, 10, 6, 7, 2, 1, 6, 6, 2, 4, 14, 15, 16, 4, 19, 3, 19, 26, 5, 3, 4, 7, 6, 10, 9, 10, 20, 18, 3, 20, 10\} \]

The sorted data is:

\[ \{1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 8, 9, 10, 10, 10, 10, 10, 14, 15, 16, 18, 18, 19, 19, 20, 20, 21, 22, 23, 26, 26, 27\} \]

To find the 50th percentile (median), we use the formula for rank:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (48 + 1) = 24.0 \]

where \( N \) is the number of data points (48 in this case).

Since the rank is 24.0, we take the values at positions 24 and 25 in the sorted list:

\[ X_{\text{lower}} = 6 \quad \text{and} \quad X_{\text{upper}} = 7 \]

We calculate the 50th percentile using the averaging formula:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{6 + 7}{2} = 6.0 \]

Final Answer