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Chapter 3: Problem 17

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. Listed below in dollars are the amounts it costs for marriage proposal packages at the different Major League Baseball stadiums. Five of the teams don't allow proposals. Are there any outliers? $$\begin{array}{rrrrrrrrrrrrr} 39 & 50 & 50 & 50 & 55 & 55 & 75 & 85 & 100 & 115 & 175 & 175 & 200 \\\209 & 250 & 250 & 350 & 400 & 450 & 500 & 500 & 500 & 500 & 1500 & 2500 &\end{array}$$

Short Answer

Expert verified
Yes, there are outliers. They are 1500 and 2500.

Step by step solution

01

Arrange the data

First, arrange all the given prices in ascending order to make the following calculations easier.
02

Mean

Find the mean of the data set. The mean is the sum of all observations divided by the number of observations. Sum of data: 39 + 50 + 50 + 50 + 55 + 55 + 75 + 85 + 100 + 115 + 175 + 175 + 200 + 209 + 250 + 250 + 350 + 400 + 450 + 500 + 500 + 500 + 500 + 1500 + 2500Number of observations: 25Mean = \(\frac{\sum x_i}{n}\)
03

Median

Find the median of the data set. The median is the middle value when the data is arranged in order.Since there are 25 observations (an odd number), the median is the 13th value.Median: Check the 13th value in the arranged data.
04

Mode

Determine the mode of the data set. The mode is the value that occurs most frequently. Check the frequency of each value to find the mode.
05

Midrange

Calculate the midrange. The midrange is the average of the smallest and largest values in the data set. Midrange = \(\frac{\text{Minimum value} + \text{Maximum value}}{2}\)
06

Identify outliers

Use the interquartile range (IQR) to identify outliers:1. Find the first quartile (Q1) and the third quartile (Q3).2. Calculate the IQR: IQR = Q3 - Q1.3. Determine the lower and upper bounds using: Lower bound = Q1 - 1.5 * IQR, Upper bound = Q3 + 1.5 * IQR.4. Any data points outside these bounds are considered outliers.
07

Answer the question

Examine the data points using the bounds found in the previous step to determine if there are any outliers.

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Key Concepts

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

Mean Calculation
The mean, also known as the average, is a way to measure the central tendency of a data set. To calculate the mean, you sum up all the individual data points and then divide by the number of data points.
In the provided exercise, we have:
  • Data Points: 25
  • Sum of Data: 13718

The formula for the mean is: \( \text{Mean} = \frac{ \text{Sum of Data Points} }{ \text{Number of Data Points} } \)
Using the values from the exercise, we get: \( \text{Mean} = \frac{13718}{25} = 548.72 \)
Median Calculation
The median is the middle value in a list when the numbers are arranged in order. It gives a good indication of the central value of a data set, especially when dealing with skewed distributions.
For the given exercise with 25 data points, the median is the 13th value in the ascending order list. This is because we use the formula: \( \text{Median} = \text{Value at} \frac{n+1}{2} \text{position} \)
After arranging the data in ascending order, the 13th value is 200. So, the median is 200.
Mode Calculation
The mode represents the most frequently occurring value(s) in a data set. Sometimes, a data set can have one mode, more than one mode, or no mode at all.
In the provided exercise, we look at the frequency of each number. The values that repeat most often are 50 and 500, each appearing 4 times.
Therefore, the data set has two modes: 50 and 500.
Midrange Calculation
The midrange is a measure of central tendency calculated by averaging the highest and the lowest values in the dataset. It can provide a quick overview of the range midpoint, but can be heavily influenced by outliers.
The formula for the midrange is: \( \text{Midrange} = \frac{\text{Minimum Value} + \text{Maximum Value}}{2} \).
Using the values from the data set:
  • Minimum Value: 39
  • Maximum Value: 2500
The midrange is: \( \text{Midrange} = \frac{39 + 2500}{2} = 1269.5 \)
Outlier Detection
Outliers are data points that are significantly different from the majority of a data set. Detecting outliers is crucial because they can affect the results of data analysis.
To identify outliers, we use the Interquartile Range (IQR) method. The steps involve:
  • Calculating the first quartile (Q1) and third quartile (Q3).
  • Finding the IQR: \( \text{IQR} = \text{Q3} - \text{Q1} \).
  • Determining the lower and upper bounds:
  • Lower Bound: \( \text{Q1} - 1.5 * \text{IQR} \)
  • Upper Bound: \( \text{Q3} + 1.5 * \text{IQR} \)
Any data points outside these bounds are considered outliers.
Interquartile Range (IQR)
The Interquartile Range (IQR) measures the spread of the middle 50% of the data. It is a robust measure of variability that is not affected by outliers, making it a reliable measure of spread.
To calculate the IQR:
  • Find the first quartile (Q1), which is the median of the lower half of the data.
  • Find the third quartile (Q3), which is the median of the upper half of the data.
  • IQR is then given as: \( \text{IQR} = \text{Q3} - \text{Q1} \)
Knowing the IQR helps in identifying and managing outliers, ensuring that analyses are not skewed by exceptionally high or low values.

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Most popular questions from this chapter

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. Listed below are measured amounts of caffeine (mg per 12 oz of drink) obtained in one can from each of 20 brands (7-UP, A\&W Root Beer, Cherry Coke, .. Tab). Are the statistics representative of the population of all cans of the same 20 brands consumed by Americans? $$\begin{array}{rrrrrrrrrr}0 & 0 & 34 & 34 & 34 & 45 & 41 & 51 & 55 & 36 & 47 & 41 & 0 & 0 & 53 & 54 & 38 & 0 & 41 & 47\end{array}$$

Use the following cell phone airport data speeds (Mbps) from Sprint. Find the percentile corresponding to the given data speed. $$\begin{array}{cccccccccc} 0.2 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.3 & 0.4 & 0.4 & 0.4 \\ 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.6 & 0.6 & 0.7 & 0.8 & 1.0 \\ 1.1 & 1.1 & 1.2 & 1.2 & 1.6 & 1.6 & 2.1 & 2.1 & 2.3 & 2.4 \\ 2.5 & 2.7 & 2.7 & 2.7 & 3.2 & 3.4 & 3.6 & 3.8 & 4.0 & 4.0 \\ 5.0 & 5.6 & 8.2 & 9.6 & 10.6 & 13.0 & 14.1 & 15.1 & 15.2 & 30.4 \end{array}$$ $$13.0 \mathrm{Mbps}$$

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. Listed below are annual U.S. sales of vinyl record albums (millions of units). The numbers of albums sold are listed in chronological order, and the last entry represents the most recent year. Do the measures of center give us any information about a changing trend over time? \(\begin{array}{rrrrrrr}0.3 & 0.6 & 0.8 & 1.1 & 1.1 & 1.4 & 1.4 & 1.5 & 1.2 & 1.3 & 1.4 & 1.2 & 0.9 & 0.9\end{array}$$\begin{array}{rrrrrr}1 & 1.9 & 2.5 & 2.8 & 3.9 & 4.6 & 6.1\end{array}\)

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. In a study of speed dating conducted at Columbia University, female subjects were asked to rate the attractiveness of their male dates, and a sample of the results is listed below (1 = not attractive; 10 = extremely attractive). Can the results be used to describe the attractiveness of the population of adult males? $$583861037985568873556878887$$

Express all z scores with two decimal places. For the Verizon airport data speeds (Mbps) listed in Data Set 32 "Airport Data Speeds" in Appendix B, the highest speed of 77.8 Mbps was measured at Atlanta's (ATL) international airport. The complete list of 50 Verizon data speeds has a mean of \(\bar{x}=17.60\) Mbps and a standard deviation of \(s=16.02 \mathrm{Mbps.}\) a. What is the difference between Verizon's data speed at Atlanta's international airport and the mean of all of Verizon's data speeds? b. How many standard deviations is that [the difference found in part (a)]? c. Convert Verizon's data speed at Atlanta's international airport to a \(z\) score. d. If we consider data speeds that convert to \(z\) scores between -2 and 2 to be neither significantly low nor significantly high, is Verizon's speed at Atllanta significant?
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