Sure, let’s work through the problem step-by-step.
### Part a) Finding the Five-Number Summary
The five-number summary consists of five values: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
1. Minimum Value:
- The smallest number in the list.
- In our case, the minimum value is 27.
2. First Quartile (Q1):
- This is the median of the lower half of the dataset (not including the overall median if the number of data points is odd).
- For the dataset [27, 28, 29, 35, 35, 36, 37, 42, 43, 46, 79], the lower half (before the median 36) is [27, 28, 29, 35, 35].
- The median of [27, 28, 29, 35, 35] is 29.
- Therefore, Q1 is 32.0.
3. Median:
- The middle value of the data set.
- In this dataset with 11 values, the median is the 6th value, which is 36.0.
4. Third Quartile (Q3):
- This is the median of the upper half of the dataset (not including the overall median if the number of data points is odd).
- For the dataset [27, 28, 29, 35, 35, 36, 37, 42, 43, 46, 79], the upper half (after the median 36) is [37, 42, 43, 46, 79].
- The median of [37, 42, 43, 46, 79] is 43.
- Therefore, Q3 is 42.5.
5. Maximum Value:
- The largest number in the list.
- In our case, the maximum value is 79.
So, the five-number summary is:
- Minimum: 27
- Q1: 32.0
- Median: 36.0
- Q3: 42.5
- Maximum: 79
### Part b) Box Plot
Using the five-number summary from part (a), we can display the data on a box plot.
Steps to Create a Box Plot:
1. Draw a number line that accommodates the range of the data.
2. Mark the five-number summary values on the number line.
3. Draw a rectangular box from Q1 (32.0) to Q3 (42.5).
4. Draw a line inside the box at the median value (36.0).
5. Draw "whiskers" from the minimum value (27) to Q1 (32.0) and from Q3 (42.5) to the maximum value (79).
You can sketch this as follows:
```
|----------|-------|--------|-------|-----|
27 32.0 36.0 42.5 79
```
This visual representation helps to understand the distribution of the data and identify any potential outliers. For instance, 79 might be considered an outlier as it's significantly higher than other values in this dataset.
