Question

the table below shows the highest temperatures recorded in august for several one town.
year temperature (°f)
1990 86
1991 78
1992 84
1993 98
1994 81
1995 77
1996 88
1997 93
the interquartile range of these data is (1) 7 (2) 10 (3) 11 (4) 18

1. given the following data set: 65,70,70,70,70,80,80,80,85,90,90,95,95,95,100

which representations are correct for this data set?

1. i and ii
2. i and iii only
3. ii and iii only
4. i, ii, and iii

Explanation:

Step1: Find the inter - quartile range for the first data set

First, order the temperature data: $77,78,81,84,86,88,93,98$. There are $n = 8$ data points.
The median ($Q_2$) is the average of the 4th and 5th ordered values. So, $Q_2=\frac{84 + 86}{2}=85$.
The lower half of the data is $77,78,81,84$. The median of the lower half ($Q_1$) is the average of the 2nd and 3rd values, $Q_1=\frac{78+81}{2}=79.5$.
The upper half of the data is $86,88,93,98$. The median of the upper half ($Q_3$) is the average of the 2nd and 3rd values of the upper - half, $Q_3=\frac{88 + 93}{2}=90.5$.
The inter - quartile range $IQR=Q_3 - Q_1=90.5-79.5 = 11$.

Step2: Analyze the second data set representations

• For the dot - plot (I): It correctly represents the frequency of each data value in the set $65,70,70,70,70,80,80,80,85,90,90,95,95,95,100$.
• For the box - plot (II): First, order the data. $n = 15$. The median ($Q_2$) is the 8th value, which is $80$. The lower half of the data has $n_1=7$ values, and its median ($Q_1$) is the 4th value, which is $70$. The upper half of the data has $n_2 = 7$ values, and its median ($Q_3$) is the 12th value, which is $90$. The box - plot is drawn correctly with the minimum value $65$, $Q_1 = 70$, $Q_2=80$, $Q_3 = 90$, and maximum value $100$.
• For the histogram (III): If we group the data into intervals, for example, $61 - 71$, $71 - 81$, $81 - 91$, $91 - 101$, the frequencies are calculated correctly based on the data set.

Answer:

1. 11
2. 4) I, II, and III