Question

the table below shows the highest temperatures recorded in august for several one town.
the inter - quartile 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: For the first problem (finding inter - quartile range)

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. $Q_2=\frac{84 + 86}{2}=85$. The lower half of the data is $77,78,81,84$, and its median ($Q_1$) is $\frac{78+81}{2}=79.5$. The upper half of the data is $86,88,93,98$, and its median ($Q_3$) is $\frac{88 + 93}{2}=90.5$. The inter - quartile range $IQR=Q_3 - Q_1=90.5-79.5 = 11$.

Step2: For the second problem (data representation)

• Dot - plot (I): Each data value is represented by a dot. In the given dot - plot, the frequencies of the values match the data set $65,70,70,70,70,80,80,80,85,90,90,95,95,95,100$. For example, there is 1 dot at 65, 4 dots at 70, etc.
• Box - plot (II): First, find the five - number summary. The minimum is 65, $Q_1$ (for $n = 15$ data points, the position of $Q_1$ is $\frac{15 + 1}{4}=4$th value, so $Q_1 = 70$), the median ($Q_2$) is the 8th value, so $Q_2=80$, $Q_3$ (position is $3\times\frac{15 + 1}{4}=12$th value, so $Q_3 = 95$), and the maximum is 100. The box - plot is drawn correctly with the box from $Q_1$ to $Q_3$ and the median line inside the box.
• Histogram (III): If we assume appropriate class intervals, the frequencies in each class interval match the distribution of the data set. For example, if we have class intervals like $61 - 71$, $71 - 81$, $81 - 91$, $91 - 101$, the frequencies in each interval can be counted from the data set and match the histogram.

Answer:

1. (3) 11
2. 4) I, II, and III