UpStudy Free Solution:

To construct a boxplot and identify the 5-number summary for the given systolic readings, we need to follow these steps:

1. Organize the data in ascending order:

\[120, 120, 122, 125, 126, 128, 129, 130, 131, 135, 137, 140, 149, 150\]

2. Identify the 5-number summary:

- Minimum (Min): The smallest value in the dataset.

- First Quartile (Q1): The median of the lower half of the dataset (excluding the median if the number of observations is odd).

- Median (Q2): The middle value of the dataset.

- Third Quartile (Q3): The median of the upper half of the dataset (excluding the median if the number of observations is odd).

- Maximum (Max): The largest value in the dataset.

Let's calculate these:

- Minimum (Min): 120

- First Quartile (Q1): The median of the first half (120, 120, 122, 125, 126, 128, 129):

\[Q1 = 125\]

- Median (Q2): The median of the entire dataset:

\[Q2 = 129.5\]

- Third Quartile (Q3): The median of the second half (130, 131, 135, 137, 140, 149, 150):

\[Q3 = 137\]

- Maximum (Max): 150

So, the 5-number summary is:

\[\text { Min} = 120, \ Q1 = 125, \ Q2 = 129.5, \ Q3 = 137, \ \text { Max} = 150\]

3. Identify outliers using the IQR (Interquartile Range):

- Calculate the IQR:

\[IQR = Q3 - Q1 = 137 - 125 = 12\]

- Determine the lower and upper bounds for outliers:

\[\text { Lower bound} = Q1 - 1.5 \times IQR = 125 - 1.5 \times 12 = 125 - 18 = 107\]

\[\text { Upper bound} = Q3 + 1.5 \times IQR = 137 + 1.5 \times 12 = 137 + 18 = 155\]

Any value below 107 or above 155 is considered an outlier. In this dataset:

- The minimum value is 120, which is above 107.

- The maximum value is 150, which is below 155.

Therefore, there are no outliers in this dataset.

4. Construct the boxplot:

- Draw a number line that includes the range of the data (from 120 to 150).

- Draw a box from Q1 (125) to Q3 (137).

- Draw a line inside the box at the median (129.5).

- Draw "whiskers" from the minimum (120) to Q1 (125) and from Q3 (137) to the maximum (150).

The boxplot visually represents the distribution of the data and confirms there are no outliers based on the IQR method.

Supplemental Knowledge

Understanding how to construct a boxplot and identify the 5-number summary is essential in descriptive statistics. Here’s a detailed explanation of these concepts:

1. 5-Number Summary:

- The 5-number summary provides a quick overview of a dataset's distribution.

- It includes the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

2. Steps to Calculate the 5-Number Summary:

- Minimum (Min): The smallest value in the dataset.

- First Quartile (Q1): The median of the lower half of the dataset.

- Median (Q2): The middle value when the data is ordered.

- Third Quartile (Q3): The median of the upper half of the dataset.

- Maximum (Max): The largest value in the dataset.

3. Interquartile Range (IQR) and Outliers:

- IQR is calculated as \(Q3 - Q1\).

- Outliers are identified using:

\[\text { Lower bound} = Q1 - 1.5 \times IQR\]

\[\text { Upper bound} = Q3 + 1.5 \times IQR\]

- Values outside these bounds are considered outliers.

4. Constructing a Boxplot:

- A boxplot visually represents the distribution of data based on its quartiles.

- It consists of:

- A central box from Q1 to Q3.

- A line inside the box at Q2 (the median).

- Whiskers extending from Min to Q1 and from Q3 to Max.

Mastering statistical tools like boxplots and five-number summaries is key to developing your analytical abilities in various fields such as research or data science. If you need further explanation or help tackling complex statistics issues or any other challenging topics - UpStudy has got your covered!