Finding and Interprating Parcentilgs In Exercises 37-40, use \( t \) data set, which represents wait times (in minutes) for various services at state's Department of Motor Vehicles Locations. \[ \begin{array}{rrrrrrrrrrrrrrrrrr} 6 & 10 & 1 & 22 & 23 & 10 & 6 & 7 & 2 & 1 & 6 & 6 & 2 & 4 & 14 & 15 & 16 & 4 \\ 19 & 3 & 19 & 26 & 5 & 3 & 4 & 7 & 6 & 10 & 9 & 10 & 20 & 18 & 3 & 20 & 10 & 13 \\ 14 & 11 & 14 & 17 & 4 & 27 & 4 & 8 & 4 & 3 & 26 & 18 & 21 & 1 & 3 & 3 & 5 & 5 \end{array} \] 37. Draw an ogive to show corresponding percentiles for the data. 38. Which wait time represents the 50th pereentile? How would you interpret 30. Find the pereentile that corresponds to a wait time of 20 minutes. 40. Which wait times are between the 25 th and 75 th percentiles?

To solve the problem, we will follow these steps: 1. **Organize the data**: We will first sort the wait times in ascending order. 2. **Calculate the percentiles**: We will find the 25th, 50th, and 75th percentiles, as well as the percentile corresponding to a wait time of 20 minutes. 3. **Draw an ogive**: We will create an ogive to visualize the percentiles. ### Step 1: Organize the Data The given wait times are: \[ \begin{array}{rrrrrrrrrrrrrrrrrr} 6 & 10 & 1 & 22 & 23 & 10 & 6 & 7 & 2 & 1 & 6 & 6 & 2 & 4 & 14 & 15 & 16 & 4 \\ 19 & 3 & 19 & 26 & 5 & 3 & 4 & 7 & 6 & 10 & 9 & 10 & 20 & 18 & 3 & 20 & 10 & 13 \\ 14 & 11 & 14 & 17 & 4 & 27 & 4 & 8 & 4 & 3 & 26 & 18 & 21 & 1 & 3 & 3 & 5 & 5 \end{array} \] Let's sort this data in ascending order. ### Step 2: Sort the Data Sorting the data gives us: \[ 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 9, 10, 10, 10, 10, 10, 11, 14, 14, 14, 15, 16, 17, 18, 18, 19, 19, 20, 20, 21, 22, 23, 26, 26, 27 \] ### Step 3: Calculate the Percentiles To find the percentiles, we will use the formula for the \( k \)-th percentile: \[ P_k = \frac{k}{100} \times (n + 1) \] where \( n \) is the number of data points. 1. **Total number of data points**: \( n = 50 \) 2. **50th Percentile (Median)**: \[ P_{50} = \frac{50}{100} \times (50 + 1) = 25.5 \] The 50th percentile is the average of the 25th and 26th values in the sorted list. 3. **25th Percentile**: \[ P_{25} = \frac{25}{100} \times (50 + 1) = 12.75 \] The 25th percentile is the average of the 12th and 13th values. 4. **75th Percentile**: \[ P_{75} = \frac{75}{100} \times (50 + 1) = 38.25 \] The 75th percentile is the average of the 38th and 39th values. 5. **Percentile for 20 minutes**: We will find the position of 20 minutes in the sorted data to determine its percentile. ### Step 4: Calculate Percentiles Now, let's calculate the specific values for the 25th, 50th, and 75th percentiles, as well as the percentile for 20 minutes. #### Finding the Percentiles - **50th Percentile**: - 25th value: 6 - 26th value: 7 - \( P_{50} = \frac{6 + 7}{2} = 6.5 \) - **25th Percentile**: - 12th value: 3 - 13th value: 4 - \( P_{25} = \frac{3 + 4}{2} = 3.5 \) - **75th Percentile**: - 38th value: 19 - 39th value: 20 - \( P_{75} = \frac{19 + 20}{2} = 19.5 \) - **Percentile for 20 minutes**: - Count the number of values less than or equal to 20 minutes. - There are 43 values less than or equal to 20 minutes. - Percentile = \( \frac{43}{50} \times 100 = 86\% \) ### Step 5: Ogive To draw an ogive, we will plot the cumulative frequency against the wait times. The cumulative frequency is calculated by counting how many data points are less than or equal to each unique wait time. ### Summary of Results - **50th Percentile**: 6.5 minutes (Median) - **25th Percentile**: 3.5 minutes - **75th Percentile**: 19.5 minutes - **Percentile for 20 minutes**: 86th percentile ### Wait Times Between 25th and 75th Percentiles The wait times between the 25th and 75th percentiles (3.5 and 19.5 minutes) are: \[ 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 9, 10, 10, 10, 10, 10, 11, 14, 14, 14, 15, 16, 17, 18, 18, 19 \] This concludes the solution to the problem. If you need further assistance with the ogive or any other calculations, please let me know!