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Chapter 3: Problem 28

A survey of 40 randomly selected full-time Joliet Junior College students was conducted in the Fall 2015 semester. In the survey, the students were asked to disclose their weekly spending on entertainment. The results of the survey are as follows: $$ \begin{array}{rrrrrrrr} \hline 21 & 54 & 64 & 33 & 65 & 32 & 21 & 16 \\ \hline 22 & 39 & 67 & 54 & 22 & 51 & 26 & 14 \\ \hline 115 & 7 & 80 & 59 & 20 & 33 & 13 & 36 \\ \hline 36 & 10 & 12 & 101 & 1000 & 26 & 38 & 8 \\ \hline 28 & 28 & 75 & 50 & 27 & 35 & 9 & 48 \\ \hline \end{array} $$ (a) Check the data set for outliers. (b) Draw a histogram of the data and label the outliers on the histogram. (c) Provide an explanation for the outliers.

Short Answer

Expert verified
The data point 1000 is an outlier. The histogram will show a typical range with an anomaly at 1000.

Step by step solution

01

Organize the Data

Sort the data in ascending order: 7, 8, 9, 10, 12, 13, 14, 16, 20, 21, 21, 22, 22, 26, 26, 27, 28, 28, 32, 33, 33, 35, 36, 36, 38, 39, 48, 50, 51, 54, 54, 59, 64, 65, 67, 75, 80, 101, 115, 1000.
02

Calculate the Quartiles

Identify the first quartile (Q1) and third quartile (Q3). For this dataset, Q1 is the value of the data point at the 10th percentile, and Q3 is the value of the data point at the 30th percentile. Using the formula for position: Q1 position = 0.25(n + 1) = 0.25(40 + 1) = 10.25, Q3 position = 0.75(n + 1) = 0.75(40 + 1) = 30.75. So, Q1 = 21.75 and Q3 = 59.25.
03

Calculate the Interquartile Range (IQR)

Subtract Q1 from Q3 to get the Interquartile Range (IQR): IQR = Q3 - Q1 = 59.25 - 21.75 = 37.5.
04

Identify the Outliers

Use the IQR to determine the boundaries for outliers: Lower boundary: Q1 - 1.5(IQR) = 21.75 - 1.5(37.5) = -34.5, Upper boundary: Q3 + 1.5(IQR) = 59.25 + 1.5(37.5) = 115.5. Any data point below -34.5 or above 115.5 is considered an outlier. In this case, 1000 is an outlier.
05

Draw the Histogram

Create a histogram of the data using bins (e.g., 0-100, 101-200, ..., 901-1000). Label the outliers, particularly highlighting the data point 1000.
06

Explain the Outliers

Explain why 1000 is an outlier. This value could be due to an error in data entry or a special circumstance causing unusually high spending that does not represent the typical student spending pattern.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quartiles
Quartiles help in understanding the distribution of a dataset by dividing it into four equal parts.
The three main quartiles to consider are Q1 (first quartile), Q2 (median or second quartile), and Q3 (third quartile).
Quartiles are useful in identifying the spread and center of the data.
Q1 represents the 25th percentile. This means it is the value below which 25% of the data fall.
Q3 is the 75th percentile, showing the value below which 75% of the data points are.
In the given exercise, sorting the data and then calculating the quartiles was the first and critical step.
For the Joliet Junior College students鈥� weekly spending, Q1 was calculated to be 21.75, and Q3 was 59.25.
This helped in later calculations to determine outliers.
interquartile range
Interquartile Range (IQR) measures the spread of the middle 50% of data points.
It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
The formula is: \[ IQR = Q3 - Q1 \]
The IQR is useful for identifying variability within a dataset as it ignores extreme values or outliers.
In our dataset, the IQR was calculated to be 37.5, indicative of a moderate spread of weekly spending values.
IQR is then used to identify outliers by applying it in a standard rule: any point below \[ Q1 - 1.5(IQR) \] or above \[ Q3 + 1.5(IQR) \]
This step was crucial in identifying 1000 as a significant outlier in the dataset.
histogram
A histogram provides a visual representation of the distribution of a dataset.
It divides data into bins or intervals, showing the frequency of data points in each bin.
By plotting the data, students can quickly see patterns such as the spread and central tendency.
In this exercise, creating a histogram involves dividing the weekly spending amounts into appropriate bins such as 0-100, 101-200, etc.
This plot can help identify the data distribution and highlight outliers like the value 1000.
When interpreting histograms, look for skewness and central clustering of data points.
A properly labeled histogram, noting the outlier, provides a clear visual understanding of the dataset's characteristics.

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