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What percentage of the general U.S. population have bachelor's degrees? The Statistical Abstract of the United States, 120 th Edition, gives the percentage of bachelor's degrees by state. For convenience, the data are sorted in increasing order. \(\begin{array}{llllllllll}17 & 18 & 18 & 18 & 19 & 20 & 20 & 20 & 21 & 21 \\ 21 & 21 & 22 & 22 & 22 & 22 & 22 & 22 & 23 & 23 \\\ 24 & 24 & 24 & 24 & 24 & 24 & 24 & 24 & 25 & 26 \\ 26 & 26 & 26 & 26 & 26 & 27 & 27 & 27 & 27 & 27 \\ 28 & 28 & 29 & 31 & 31 & 32 & 32 & 34 & 35 & 38\end{array}\) (a) Make a box-and-whisker plot and find the interquartile range. (b) Illinois has a bachelor's degree percentage rate of about \(26 \%\). Into what quartile does this rate fall?

Step by step solution

01

Organize the Data

The provided data needs to be ordered to make it easier to identify the quartiles. The data set is \[17, 18, 18, 18, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 23, 23,\]\[24, 24, 24, 24, 24, 24, 24, 24, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 29, 31, 31, 32, 32, 34, 35, 38\].Check this sequence to verify that it is in increasing order.

02

Determine the Quartiles

To create a box-and-whisker plot, we need to determine the quartiles. First, find the median (Q2), which is the middle value:- With 50 states, the median is between the 25th and 26th values.- Median = \(\frac{24 + 24}{2} = 24\).Next, find Q1 and Q3:- Q1 is the median of the first half (values 1 to 25): median is 22.- Q3 is the median of the second half (values 26 to 50): median is 27.

03

Calculate the Interquartile Range (IQR)

The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1):\[IQR = Q3 - Q1 = 27 - 22 = 5\].

04

Create the Box-and-Whisker Plot

To make the box-and-whisker plot: - Draw a number line that covers the data range. - Identify minimum (17), Q1 (22), median (24), Q3 (27), and maximum (38). - Draw a box from Q1 to Q3 with a line at the median. - Extend "whiskers" from the box to the minimum and maximum values.

05

Determine the Quartile for Illinois

Illinois has a bachelor's degree percentage rate of 26%. - Since 26% is greater than Q1 (22) but not greater than Q3 (27), it falls into the third quartile. - Values between Q2 (24) and Q3 (27) are in the third quartile, and 26 falls in this range.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interquartile Range

The interquartile range, or IQR, is a measure of statistical dispersion. It tells us how spread out the middle 50% of values in a data set are. To find it, we take the difference between the third quartile (Q3) and the first quartile (Q1). In simpler terms, the IQR shows the spread of the central chunk of your data.

Here's a quick guide on how to calculate it:

• Identify Q1 (the 25th percentile) and Q3 (the 75th percentile) of your data set.
• Subtract Q1 from Q3: \[ IQR = Q3 - Q1 \]

The IQR is valuable because it helps us understand how concentrated our data is around the median, and it can also be used to identify outliers.

In our exercise, the data's IQR is 5, calculated as \(27 - 22 = 5 \).This means that the range of the middle 50% of states, in terms of bachelor's degree rates, covers only 5 percentage points.

Quartiles

Quartiles are numerical values that divide your data set into quarters, providing a clear view of its distribution. They are composed of three points: Q1, Q2 (median), and Q3. Each quartile represents a specific portion of the data:

• Q1 (First Quartile): This is the midpoint of the first half of the data. It separates the lowest 25% of the data from the rest.
• Q2 (Median): Also known as the second quartile, it splits the data into two equal halves. This is explained in more detail below.
• Q3 (Third Quartile): The midpoint of the upper half. It separates the highest 25% from the lowest 75% of the data.

Quartiles provide valuable insights, such as understanding central tendency and variability—key statistical concepts that help summarize large data sets. In our exercise, these quartiles for the bachelor's degree data are Q1 = 22, Q2 = 24, and Q3 = 27. By analyzing these, you can see how data is clustered around the median and where it spreads out.

Median

The median is a central value in a data set and acts as an essential measure of central tendency. It represents the middle number when all values are organized in increasing order. Unlike the mean, the median is unaffected by extremely high or low values, making it a reliable statistic in skewed distributions.

To find the median in a set with an even number of values, average the two numbers in the middle. Here's how you do it:

• Sort your data set in increasing order.
• Count the data points. If there’s an odd number of data points, the median is the middle number.
• If even, take the average of the two middle numbers.

In our example, the data set has an even number of 50 states. Therefore, the median, or Q2, is found by averaging the 25th and 26th values:\[ \text{Median} = \frac{24 + 24}{2} = 24 \]The median offers a central reference point for the data and helps compare halves of sorted data.