91Ó°ÊÓ

The following table shows the gas tax (in cents per gallon) in each of the southern U.S. states. (Source: 2017 World Almanac and Book of Facts) a. Find and interpret the median gas tax using a sentence in context. b. Find and interpret the interquartile range. c. What is the mean gas tax? d. Note that the mean for this data set is greater than the median. What does this indicate about the shape of the data? Make a graph of the data and discuss the shape of the data. $$ \begin{array}{|l|l|} \hline \text { State } & \begin{array}{c} \text { Gas Tax } \\ \text { (cents/gallon) } \end{array} \\ \hline \text { Alabama } & 39.3 \\ \hline \text { Arkansas } & 40.2 \\ \hline \text { Delaware } & 41.4 \\ \hline \text { District of } & \\ \text { Columbia } & 41.9 \\ \hline \text { Florida } & 55 \\ \hline \text { Georgia } & 49.4 \\ \hline \text { Kentucky } & 44.4 \\ \hline \text { Louisiana } & 38.4 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \text { State } & \begin{array}{c} \text { Gas Tax } \\ \text { (cents/gallon) } \end{array} \\ \hline \text { Maryland } & 51 \\ \hline \text { Mississippi } & 37.2 \\ \hline \text { N. Carolina } & 53.7 \\ \hline \text { S. Carolina } & 35.2 \\ \hline \text { Tennessee } & 39.8 \\ \hline \text { Texas } & 38.4 \\ \hline \text { Virginia } & 40.7 \\ \hline \text { W. Virginia } & 51.6 \\ \hline \end{array} $$

Step by step solution

01

Calculate the Median

Order the gas tax values from smallest to largest. Since there are 17 states, the median is the value in the 9th position, because an odd number of observations means the median is the middle one. So, the median is 41.4 cents.

02

Interpret the Median

The median being 41.4 cents means that half of the southern U.S. states have a gas tax under 41.4 cents per gallon, and half have a gas tax over 41.4 cents per gallon.

03

Calculate the Interquartile Range

This represents the middle 50% of the data. The first quartile (Q1) is the median of the first half of the data, and the third quartile (Q3) is the median of the second half. The interquartile range is Q3 - Q1, so calculate these values to find it.

04

Interpret the Interquartile Range

Interpreting an interquartile range typically involves explaining the spread of the middle 50% of values in the dataset. The larger it is, the more spread out these values are.

05

Calculate the Mean

The mean (or average) gas tax is calculated by adding up all the values and then dividing by the number of states. Calculate this to find the mean gas tax.

06

Interpret the Mean-Median Relationship and Data Shape

If the mean is greater than the median, the data is positively skewed; this is because the mean is influenced by high values more than the median is. Graph the data and confirm the shape.

07

Discuss the Shape of the Data

Based on the shape of the graph, discuss the distribution of the data. If it is skewed to the right, confirm that this matches your earlier interpretation of the mean-median relationship.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Median

The median is the middle value of a data set when it's arranged in ascending or descending order. In the context of gas taxes among southern U.S. states, finding the median involves ordering the gas tax amounts from the lowest to the highest and identifying the value that divides the list into two equal halves.

In our case, the median gas tax is 41.4 cents per gallon. This means that if you line up all the states by their gas tax rate, Alabama having the least and West Virginia the most, the gas tax of Delaware would be right in the middle, splitting the dataset into halves. Half of the states charge less than or equal to 41.4 cents per gallon, and the other half charge more.

This measure is particularly useful because it is not swayed by extreme values or outliers that could skew the data—an issue that can affect the mean, or average, of the dataset.

Interquartile Range

The interquartile range (IQR) is a measure of variability that indicates the spread of the middle 50% of data points in a dataset. It's calculated by subtracting the first quartile (the median of the lower half of the data) from the third quartile (the median of the upper half).

To find the interquartile range for the gas tax example, we first determine the first and third quartiles, which may involve splitting the data into two halves and finding their respective medians. After that, we subtract the first quartile from the third quartile to get the IQR.

A larger IQR indicates a greater spread in the middle half of the dataset, which means more variation in the gas tax among the states. It helps to understand how concentrated or dispersed the values are around the median, excluding outliers.

Mean

The mean, commonly referred to as the average, is the sum of all the values in a dataset divided by the number of values. It is a central measure of tendency that gives an overall impression of the data set.

When calculating the mean gas tax, we would add up all the gas tax values from each of the southern states and then divide by the total number of states. This gives us the average gas tax that you might anticipate across these states. However, unlike the median, the mean is sensitive to outliers. This means that states with particularly high or low gas taxes can distort the mean, making it higher or lower than the median - an important consideration in data analysis.

Data Distribution

Data distribution refers to the way in which data points are spread out or clustered together over a range of values. Different shapes of distributions can reveal trends and patterns, and provide insights into the dataset being analyzed.

In the gas tax example, an analysis of the mean and median revealed the data is positively skewed. This suggests that the distribution has a tail on the right side. Positively skewed distributions have a mean that is greater than the median, as higher values pull the mean upwards. This is reflected in the graph of the data, which would show a longer tail extending toward the higher gas tax rates. A graphical representation helps to visualize the skewness, concentrations, and outliers within the data.