Question

question
the data below are the monthly average high temperatures for new york city. what is the five-number summary?
40, 40, 48, 61, 72, 78, 84, 84, 76, 65, 54, 42
select the correct answer below:
sample minimum: 42, sample maximum: 78
q1: 45, median: 63, q3: 77
sample minimum: 40, sample maximum: 84
q1: 45, median: 63, q3: 77
sample minimum: 40, sample maximum: 42
q1: 54.5, median: 81, q3: 70.5
sample minimum: 40, sample maximum: 84
q1: 40, median: 61, q3: 76
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Explanation:

Step1: Sort the data

First, we sort the given data in ascending order: \(40, 40, 42, 48, 54, 61, 65, 72, 76, 78, 84, 84\)

Step2: Find the sample minimum and maximum

The sample minimum is the smallest value in the sorted data, which is \(40\). The sample maximum is the largest value, which is \(84\).

Step3: Find the median (second quartile, \(Q_2\))

Since there are \(n = 12\) data points (an even number), the median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+ 1)\)-th values. \(\frac{n}{2}=\frac{12}{2}=6\), so we take the average of the 6th and 7th values. The 6th value is \(61\) and the 7th value is \(65\). The median is \(\frac{61 + 65}{2}=\frac{126}{2}=63\).

Step4: Find the first quartile (\(Q_1\))

The first quartile is the median of the lower half of the data. The lower half of the data (excluding the median when \(n\) is even? Wait, actually, for \(n = 12\), the lower half is the first 6 data points: \(40, 40, 42, 48, 54, 61\). The median of these 6 data points (since \(n_{lower}=6\), even) is the average of the 3rd and 4th values. The 3rd value is \(42\) and the 4th value is \(48\). So \(Q_1=\frac{42 + 48}{2}=\frac{90}{2}=45\).

Step5: Find the third quartile (\(Q_3\))

The third quartile is the median of the upper half of the data. The upper half of the data is the last 6 data points: \(65, 72, 76, 78, 84, 84\). The median of these 6 data points is the average of the 3rd and 4th values. The 3rd value is \(76\) and the 4th value is \(78\). So \(Q_3=\frac{76+78}{2}=\frac{154}{2}=77\).

Answer:

Sample minimum: 40, Sample maximum: 84
Q1: 45, Median: 63, Q3: 77 (The corresponding option is the second one: Sample minimum: 40, Sample maximum: 84; Q1: 45, Median: 63, Q3: 77)