Final answer:
The five key values, or five-number summary, for the provided test scores are: minimum (42), lower quartile (54), median (64), upper quartile (70), and maximum (80). This summary shows the spread of the students' test completion times, from the fastest to the slowest.
Step-by-step explanation:
To find the five key values that describe the test scores collected from pencil-and-paper exams, also known as the five-number summary, we need to calculate the minimum, lower quartile (Q1), median, upper quartile (Q3), and maximum values from the given dataset: 42, 48, 52, 54, 59, 61, 61, 64, 65, 67, 68, 70, 72, 75, 80.
- Minimum value: The smallest number in the dataset is 42.
- Lower quartile (Q1): This represents the 25th percentile of the data. Since we have 15 values, the lower quartile will be the value that separates the lowest 25% of the data from the rest. It is the median of the first half of the data (when the data is ordered), which is the median of 42, 48, 52, 54, 59, 61, 61. The lower quartile is 54.
- Median: This is the middle value when data points are ordered. For our dataset, the median is the 8th value, which is 64.
- Upper quartile (Q3): This represents the 75th percentile of the data and is the median of the upper half of the data. The upper quartile in this set is the median of 65, 67, 68, 70, 72, 75, 80, which is 70.
- Maximum value: The largest number in the dataset is 80.
To interpret these values in context: the minimum value indicates the fastest test completion time, the lower quartile means that 25% of students finished the exam in 54 minutes or less, the median represents the time at which half of the students finished the exam, the upper quartile shows that 75% of students completed the exam in 70 minutes or less, and the maximum value indicates the slowest test completion time.