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Chapter 5: Problem 6

Brian looked up prices of thirteen used Chevrolet HHR 鈥渞etro鈥� trucks in the classified ads and found these prices: \(\mathrm{S8}, 500, \$ 8,500, \$ 8,500,\) \(\$ 9,900, \$ 10,800, \$ 10,800, \$ 11,000, \$ 12,500, \$ 12,500, \$ 13,000,\) \(\$ 13,000, \$ 14,500,\) and \(\$ 23,000.\) a. Make a frequency table for this data set. b. Find the mean. Round to the nearest dollar. c. Find the median. d. Find the mode. e. Find the range. f. Find the four quartiles. g. Find the interquartile range. h. Find the boundary for the upper outliers. i. Find the boundary for the lower outliers. j. How many outliers are there? k. Draw a modified box-and-whisker plot. Label it.

Short Answer

Expert verified
The solution will include the created frequency table, the calculated mean price, the identified median and mode prices, the calculated range, the identified quartiles and interquartile range, the identified boundaries for outliers, the count of outliers, and a sketched modified box-and-whisker plot. The exact results will depend on the calculations.

Step by step solution

01

Creating Frequency Table

Use the given data of prices and group them in a manner that it represents the number of occurrences or frequency of each price in the data set.
02

Calculate Mean

Find the sum of all the prices and divide it by the total number of prices. Round the result to the nearest dollar.
03

Determine Median

Sort the data in ascending order. The median is the middle value. If the data set has an even number of observations, the median is the average of the two middle values.
04

Identify Mode

The mode is the value that appears most frequently in a data set. Determine which price appears most often.
05

Determining Range

Subtract the smallest price from the largest price to find the range.
06

Find Quartiles

Quartiles are the values that split a data set into quarters. Q1 is the median of the lower half of the data, Q2 is the median of the whole data set, Q3 is the median of the upper half of the data.
07

Find Interquartile Range

The interquartile range is the range of the middle 50% of the data. Subtract Q1 from Q3.
08

Calculate Boundary for Upper & Lower Outliers

Outliers are either 1.5 times the IQR above the third quartile, or 1.5 times the IQR below the first quartile.
09

Identify Number of Outliers

An outlier is any price that is below the lower boundary or above the upper boundary. Count the number of outliers in the data set.
10

Draw Modified Box-and-Whisker Plot

With the information obtained so far, draw the modified box-and-whisker plot. The box represents Q1 to Q3, the line in the box is the median. The whiskers represent the data within 1.5 times the IQR, and any points outside this range are considered outliers.

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Key Concepts

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

Frequency Table
A frequency table is a way to organize data to show how often each value occurs. In this exercise, we have a set of used truck prices that we want to summarize in a table. To create a frequency table:
  • List all unique prices in ascending order.
  • Count how many times each price appears in the list.
This method helps in visualizing the distribution of prices at a glance. It's particularly useful in spotting the most frequent values, which can aid in identifying the mode.
Mean Calculation
Calculating the mean gives us the average price of the used trucks. To find the mean:
  • Add up all of the prices.
  • Divide the total sum by the number of prices in the data set.
For instance, with Brian's truck prices, you would sum all thirteen prices, then divide by thirteen. Rounding the result to the nearest dollar provides a more practical figure for everyday use. The mean acts as a central value, offering insight into the overall price trend.
Median
The median is the middle value of a data set when arranged in ascending order. It gives a better sense of the center of the data, especially when there are outliers. To find the median:
  • Sort the prices from lowest to highest.
  • If the number of prices is odd, the median is the middle value.
  • If even, it's the average of the two middle values.
In this example, since there are thirteen prices, the median is the seventh price in the sorted list.
Mode
The mode is the most frequently occurring value in a data set. It's helpful in identifying the most common price in the list. To determine the mode, simply look at your frequency table:
  • Identify the price that appears most often.
In some situations, there can be more than one mode, or even no mode at all, depending on the data configuration.
Range
The range provides a simple measure of data spread. It tells us the difference between the highest and lowest prices. To find the range:
  • Subtract the smallest price from the largest price.
This calculation gives an idea of the extent of price variability. A higher range indicates greater variability amongst the truck prices.
Quartiles
Quartiles divide your data set into four equal parts, helping to further analyze data distribution:
  • Q1 (First Quartile) is the median of the lower half.
  • Q2 is the overall median.
  • Q3 (Third Quartile) is the median of the upper half.
For Brian's truck prices, find these quartiles to gain deeper insights into how the prices are spread. Quartiles are instrumental in understanding the data's variability, especially when visualizing it with tools like a box-and-whisker plot.
Interquartile Range
The interquartile range (IQR) measures the spread of the middle 50% of a data set. It is a robust indicator of variability as it is less affected by outliers:
  • Calculate IQR by subtracting Q1 from Q3.
In the context of Brian's truck prices, a smaller IQR indicates that prices are closely packed, while a larger IQR suggests more variability among the middle prices. The IQR is also crucial in defining outlier thresholds.
Box-and-Whisker Plot
A box-and-whisker plot is a graphical representation of a data set that shows the data's distribution based on a five-number summary (minimum, Q1, median, Q3, maximum). To draw this plot:
  • Mark the positions of Q1, median, and Q3 with a box.
  • Draw horizontal lines (whiskers) from the box to the smallest and largest values within 1.5 times the IQR.
  • Any data points outside these limits are outliers, marked individually.
This plot provides a visual summary of data variability and central tendency, making it easier to interpret complex data like Brian's used truck prices.

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Most popular questions from this chapter

Leslie placed this ad in the Collector Car Monthly. 1957 Chevrolet Nomad station wagon. Tropical Turquoise, 6 cyl. auto, PS, PW AM/FM, repainted, rebuilt transmission, restored two-tone interior. Mint! Moving, sacrifice, \(\$ 52,900.555-4231\) a. If the newspaper charges \(\$ 48\) for the first three lines and \(\$ 5\) for each extra line, how much will this ad cost Leslie? b. Ruth buys the car for 8\(\%\) less than the advertised price. How much does she pay? Ruth must pay her state 6\(\%\) sales tax on the sale. How much must she pay in sales tax? c. Ruth must pay her state 6\(\%\) sales tax on the sale. How much must she pay in sales tax?

A local Penny saver charges \(\$ 11\) for each of the first three lines of a classified ad, and \(\$ 5\) for each additional line. a. What is the price of a two-line ad? b. What is the price of a five-line ad? c. If \(x\) is the number of lines in the ad, express the cost \(c(x)\) of the ad as a piece wise function.

The quartiles of a data set are \(\mathrm{Q}_{1}=50, \mathrm{Q}_{2}=72, \mathrm{Q}_{3}=110,\) and \(\mathrm{Q}_{4}=140 .\) Find the interquartile range.

The piece wise function describes a newspaper鈥檚 classified ad rates. $$y=\left\\{\begin{array}{ll}{21.50} & {\text { when } x \leq 3} \\\ {21.50+5(x-3)} & {\text { when } x>3}\end{array}\right.$$ a. If \(x\) represents the number of lines, and \(y\) represents the cost, translate the function into words. b. If the function is graphed, what are the coordinates of the cusp?

Lenny's car gets approximately 20 miles per gallon. He is planning a 750 -mile trip. a. About how many gallons of gas should Lenny plan to buy? b. At an average price of \(\$ 4.10\) per gallon, how much should Lenny expect to spend for gas?
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