Question

probability and statistics
1.6 and 1.8 assessment review
this question has four parts.
a coach recorded the number of goals scored by a soccer team
in each of its last ten games. the data are shown
3, 2, 10, 2, 1, 5, 3, 2, 1, 5
part a
what are the first quartile, the median, and the third quartile of the data?
enter your answers in the boxes.
first quartile =
median =
third quartile =
part b
what is the interquartile range of the data? show or explain how you got your answer
enter your answer and your work or explanation in the space provided
symbols
relations
geometry
groups

Explanation:

Step1: Sort the data

Sort the data set \(3,2,10,2,1,5,3,2,1,5\) in ascending order: \(1,1,2,2,2,3,3,5,5,10\).

Step2: Find the median

Since there are \(n = 10\) data points (even number of data points), the median \(Q_2\) is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. \(\frac{n}{2}=5\), \(\frac{n}{2}+1 = 6\). The 5th value is \(2\) and the 6th value is \(3\). So, \(Q_2=\frac{2 + 3}{2}=2.5\).

Step3: Find the first quartile \(Q_1\)

The lower - half of the data is \(1,1,2,2,2\). Since there are \(n_1=5\) (odd number of data points) in the lower - half, the first quartile \(Q_1\) is the \(\frac{n_1 + 1}{2}\)th value. \(\frac{5+1}{2}=3\)rd value. So, \(Q_1 = 2\).

Step4: Find the third quartile \(Q_3\)

The upper - half of the data is \(3,3,5,5,10\). Since there are \(n_2=5\) (odd number of data points) in the upper - half, the third quartile \(Q_3\) is the \(\frac{n_2+1}{2}\)th value. \(\frac{5 + 1}{2}=3\)rd value. So, \(Q_3=5\).

Step5: Calculate the inter - quartile range (IQR) for Part B

The inter - quartile range \(IQR=Q_3-Q_1\). Substitute \(Q_1 = 2\) and \(Q_3=5\). So, \(IQR=5 - 2=3\).

Answer:

first quartile \(=2\)
median \(=2.5\)
third quartile \(=5\)
For Part B, the inter - quartile range is \(3\) (because \(IQR = Q_3-Q_1\), and \(Q_3 = 5\), \(Q_1=2\)).