Question

Thirty-four college students were asked how much money they spent on textbooks for the current semester. Their responses are shown in the following stemplot.
1 2 3 3 4 5 5 6 7 8
2 1 2 3 4 5 6 8 8 9 9 9
3 1 2 2 7 8 9
4 1 4 5 7
5 1 3
6 2
7
8 1

Key: 1|2 = $120

a. Describe a procedure for identifying potential outliers, and use the procedure to decide whether there are outliers among the responses for the money spent on textbooks.
b. Based on the stemplot, write a few sentences describing the distribution of money spent on textbooks for the 34 students.

Answer 1

Answer:

(a) The outlier in the data is $810.

(b) The distribution of money spent on textbooks for the 34 students is right skewed.

Step-by-step explanation:

The data provided for the amount of money 34 college students spent on books is:

S = {120, 130, 130, 140, 150, 150, 160, 170, 180, 210, 220, 230, 240, 250, 260, 280, 280, 290, 290, 290, 310, 320, 320, 370, 380, 390, 410, 440, 450, 470, 510, 530, 620, 810}

(a)

An outlier of a data set is a value that is very different from the other values of a data set. It is either too large or too small.

The most common way to determine whether a data set consists of any outliers of not is,

• Data value that less than Q₁ - 1.5 IQR are outliers.
• Data values that are more than Q₃ + 1.5 IQR are outliers.

Here

Q₁ = first quartile

Q₃ = third quartile

IQR = Inter-quartile range = Q₃ - Q₁.

The first quartile is the value that is more than 25% of the data values. The first quartile is the median of the first half of the data.

Compute the value of first quartile as follows:

First half of data: {120, 130, 130, 140, 150, 150, 160, 170, 180, 210, 220, 230, 240, 250, 260, 280, 280}

There are 17 values.

The median of an odd data set is the middle value.

The middle value is: 180

The first quartile is Q₁ = 180.

The third quartile is the value that is more than 75% of the data values.

Compute the value of first quartile as follows:

Second half of data: {290, 290, 290, 310, 320, 320, 370, 380, 390, 410, 440, 450, 470, 510, 530, 620, 810}

There are 17 values.

The median of an odd data set is the middle value.

The middle value is: 390

The third quartile is Q₃ = 390.

Compute the inter-quartile range as follows:

IQR = Q₃ - Q₁

      = 390 - 180

      = 210

Compute the value of [Q₁ - 1.5 IQR] as follows:

$Q_{1}-1.5\ IQR =180-(1.5\times 210)=-135$

Compute the value of [Q₃ + 1.5 IQR] as follows:

$Q_{3}+1.5\ IQR =390+(1.5\times 210)=705$

There are no values that are less than [Q₁ - 1.5 IQR]. But there is one value that is more than [Q₃ + 1.5 IQR].

X = 810 > [Q₃ + 1.5 IQR] = 705

Thus, the outlier in the data is $810.

(b)

A distribution is known as to be skewed to the right, or positively skewed, when maximum of the data are collected on the left of the distribution.

In the stem plot above, it is shown that maximum of the data values are collected on the left of the chart. This implies that the distribution is positively skewed.

Thus, the distribution of money spent on textbooks for the 34 students is right skewed.