Chapter 1.3, Problem 112E

  (a)

To determine

Create a boxplot of the data shown.

Answer to Problem 112E

The box starts at 1^st quartile, ends at 3^rd quartile and has vertical line at the median.

Explanation of Solution

Given information:

Scores of Mrs. Liao’s students on their first statistics test:

93 93 87.5 91 94.5 72 96 95 93.5 93.5 73

82 45 88 80 86 85.5 87.5 81 78 86 89

92 91 98 85 82.5 88 94.5 43

Arrange the data values in ascending order:

43, 45, 72, 73, 78, 80, 81, 82, 82.5, 85, 85.5, 86, 87.5, 88, 88, 89, 91, 91, 92, 93, 93, 93.5, 94.5, 94.5, 95, 96, 98

We have

The number of data values is even.

The median of the sorted data set is the average of two middle values (87.5 and 88).

  $\text{med}.=\frac{87.5+88}{2}=87.75$

Now,

The 1^st quartile is the median of all data values below the median (calculated above).

However,

The data set contains 15 values below the median (calculated above).

Then

The 1^st quartile will be the 8^th data value.

Thus,

  $Q_1=82$

Now,

The 3^rd quartile is the median of all data values above the median (calculated above).

However,

The data set contains 15 values above the median (calculated above).

Then

The 3^rd quartile will be the 22^th data value.

Thus,

  $Q_3=93$

The observation more than 1.5 times the IQR above Q₃ or below Q₁ .

The difference between 3^rd and 1^st quartile gives the interquartile range ( IQR ).

Then

  $Q_3+1.5IQR=93+1.5\left(93-82\right)=109.5$

And

  $Q_1-1.5IQR=82-1.5\left(93-82\right)=65.5$

From the data values, the minimum is 43 and the maximum is 98.

Then

Note that

43 and 45 are outliers.

For boxplot:

The box has thequartiles, boundaries and a vertical line have been drawn at the median.

The whiskers represent the minimum and maximum of the data points which are not outliers.

X represents the outliers.

  

  (b)

To determine

Discuss the performance of the students on Mrs. Liao’s first statistics test.

Answer to Problem 112E

Students performed well on Mrs. Liao’s first statistics test.

Explanation of Solution

Given information:

Scores of Mrs. Liao’s students on their first statistics test:

93 93 87.5 91 94.5 72 96 95 93.5 93.5 73

82 45 88 80 86 85.5 87.5 81 78 86 89

92 91 98 85 82.5 88 94.5 43

Arrange the data values in ascending order:

43, 45, 72, 73, 78, 80, 81, 82, 82.5, 85, 85.5, 86, 87.5, 88, 88, 89, 91, 91, 92, 93, 93, 93.5, 94.5, 94.5, 95, 96, 98

We have

The number of data values is even.

The median of the sorted data set is the average of two middle values (87.5 and 88).

  $\text{med}.=\frac{87.5+88}{2}=87.75$

Now,

The 1^st quartile is the median of all data values below the median (calculated above).

However,

The data set contains 15 values below the median (calculated above).

Then

The 1^st quartile will be the 8^th data value.

Thus,

  $Q_1=82$

This implies

25% of the students have scored 82 or less.

Thus,

Remaining 75% of the students have scored more than 82.

This indicates

The students have performed very well on Mrs. Liao’s first test.