Question

here are yesterdays high temperatures (in fahrenheit) in 16 u.s. cities.
48, 52, 52, 53, 53, 62, 63, 64, 65, 66, 70, 72, 79, 80, 81, 82.
notice that the temperatures are ordered from least to greatest.
make a box-and-whisker plot for the data.

Explanation:

Step1: Find the minimum and maximum

The data set is \(48, 52, 52, 53, 53, 62, 63, 64, 65, 66, 70, 72, 79, 80, 81, 82\). The minimum value (\( \text{min} \)) is \(48\) and the maximum value (\( \text{max} \)) is \(82\).

Step2: Find the median (second quartile, \(Q_2\))

Since there are \(n = 16\) (even) data points, the median is the average of the \( \frac{n}{2} = 8^{\text{th}} \) and \( (\frac{n}{2}+1) = 9^{\text{th}} \) values. The \(8^{\text{th}}\) value is \(64\) and the \(9^{\text{th}}\) value is \(65\). So, \( Q_2=\frac{64 + 65}{2}=\frac{129}{2}=64.5 \).

Step3: Find the first quartile (\(Q_1\))

The first quartile is the median of the lower half of the data. The lower half is the first \(8\) values: \(48, 52, 52, 53, 53, 62, 63, 64\). Since \(n = 8\) (even), the median of the lower half is the average of the \(4^{\text{th}}\) and \(5^{\text{th}}\) values. The \(4^{\text{th}}\) value is \(53\) and the \(5^{\text{th}}\) value is \(53\). So, \( Q_1=\frac{53+53}{2}=53 \).

Step4: Find the third quartile (\(Q_3\))

The third quartile is the median of the upper half of the data. The upper half is the last \(8\) values: \(65, 66, 70, 72, 79, 80, 81, 82\). Since \(n = 8\) (even), the median of the upper half is the average of the \(4^{\text{th}}\) and \(5^{\text{th}}\) values. The \(4^{\text{th}}\) value is \(72\) and the \(5^{\text{th}}\) value is \(79\). So, \( Q_3=\frac{72 + 79}{2}=\frac{151}{2}=75.5 \).

Step5: Draw the box - and - whisker plot

• The whiskers extend from the minimum (\(48\)) to the maximum (\(82\)).
• The box starts at \(Q_1 = 53\), has a line at the median \(Q_2=64.5\), and ends at \(Q_3 = 75.5\).

To plot on the given number line (with marks at 45, 50, 55, 60, 65, 70, 75, 80, 85):

• Plot the minimum (\(48\)): between 45 and 50 (closer to 45).
• Plot \(Q_1 = 53\): between 50 and 55 (closer to 55? Wait, 53 is 3 units above 50 and 2 units below 55).
• Plot the median \(Q_2 = 64.5\): between 60 and 65 (64.5 is 4.5 units above 60 and 0.5 units below 65).
• Plot \(Q_3=75.5\): between 75 and 80 (75.5 is 0.5 units above 75 and 4.5 units below 80).
• Plot the maximum (\(82\)): between 80 and 85 (closer to 80).

Then draw the box from \(Q_1 = 53\) to \(Q_3=75.5\) with a vertical line at \(Q_2 = 64.5\), and draw the whiskers from \(48\) to \(Q_1\) and from \(Q_3\) to \(82\).

Answer:

The box - and - whisker plot has:

• Minimum: \(48\)
• \(Q_1\): \(53\)
• Median (\(Q_2\)): \(64.5\)
• \(Q_3\): \(75.5\)
• Maximum: \(82\)

When plotted on the given number line, the whiskers are from \(48\) to \(82\), the box is from \(53\) to \(75.5\) with a line at \(64.5\).