Question

your statistics class, made up of 16 students, earned the following scores on their semester exam, which are listed in the stem - and - leaf plot.
stem leaves
4 7
5
6 8
7 5 6 6 7 8
8 5 6 7 7 8
9 6 7 8
stems: tens
leaves: ones
4|7 = 47 points
(a) which of the scores is a possible outlier in the data?
(b) determine whether the following statement is true or false.
in the data, 29% of scores were between 70 and 79.
true
false

Explanation:

Step1: Write out all data values

The data values from the stem - and - leaf plot are 47, 68, 75, 76, 76, 77, 78, 85, 86, 87, 87, 88, 96, 97, 98.

Step2: Find quartiles

First, order the data: 47, 68, 75, 76, 76, 77, 78, 85, 86, 87, 87, 88, 96, 97, 98.
There are $n = 16$ data points.
The median $Q_2$ is the average of the 8th and 9th ordered values. So $Q_2=\frac{85 + 86}{2}=85.5$.
The lower half of the data is 47, 68, 75, 76, 76, 77, 78, 85. The median of the lower half $Q_1$ is the average of the 4th and 5th ordered values in the lower - half, so $Q_1=\frac{76+76}{2}=76$.
The upper half of the data is 86, 87, 87, 88, 96, 97, 98. The median of the upper half $Q_3$ is the average of the 4th and 5th ordered values in the upper - half, so $Q_3=\frac{88 + 96}{2}=92$.
The inter - quartile range $IQR=Q_3 - Q_1=92 - 76 = 16$.
The lower fence for outliers is $Q_1-1.5\times IQR=76-1.5\times16=76 - 24 = 52$.
The upper fence for outliers is $Q_3 + 1.5\times IQR=92+1.5\times16=92 + 24 = 116$.
The outlier is 47 as it is less than 52.

Step3: Calculate percentage of scores between 70 and 79

There are 5 scores (75, 76, 76, 77, 78) between 70 and 79.
The percentage is $\frac{5}{16}\times100\%=31.25\%
eq29\%$.

Answer:

(a) 47
(b) False