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Chapter 3: Problem 13

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? $$ \begin{array}{llllllllllllllllllll} 0 & 0 & 34 & 34 & 34 & 45 & 41 & 51 & 55 & 36 & 47 & 41 & 0 & 0 & 53 & 54 & 38 & 0 & 41 & 47 \end{array} $$

Short Answer

Expert verified
Mean: 34.21, Median: 41, Mode: 0, Midrange: 27.5. The statistics may not be fully representative of the population.

Step by step solution

01

Arrange the Data

First, arrange the caffeine amounts in ascending order: 0, 0, 0, 0, 34, 34, 34, 36, 38, 41, 41, 41, 45, 47, 47, 51, 53, 54, 55
02

Calculate the Mean

The mean is the sum of all values divided by the number of values. Sum of values: 0 + 0 + 0 + 0 + 34 + 34 + 34 + 36 + 38 + 41 + 41 + 41 + 45 + 47 + 47 + 51 + 53 + 54 + 55 = 650 Number of values: 19 Mean: \( \frac{650}{19} \approx 34.21\)
03

Calculate the Median

The median is the middle value. Since there are 19 values, which is an odd number, the median is the 10th value in the arranged list. Median: 41
04

Calculate the Mode

The mode is the value(s) that appear most frequently. In the list, the value 0 appears 4 times, and 41 appears 3 times. Mode: 0
05

Calculate the Midrange

The midrange is the average of the minimum and maximum values. Minimum value: 0 Maximum value: 55 Midrange: \( \frac{0 + 55}{2} = 27.5 \)
06

Evaluate Representativeness

To determine if the statistics are representative of the population, consider the sample size and selection bias. One can of each brand may not capture the variability between different production batches or conditions. Therefore, the statistics may not be fully representative of the population.

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Key Concepts

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

mean calculation
The mean is one of the most commonly used measures in descriptive statistics. It represents the average of a data set. To calculate the mean, you add all the values together and then divide by the number of values. For instance, in the given exercise, we first sum the caffeine amounts of all 19 drink brands: 0 + 0 + 0 + 34 + 34 + 34 + 36 + 38 + 41 + 41 + 41 + 45 + 47 + 47 + 51 + 53 + 54 + 55, which equals 650.
Next, we divide this sum by the number of values (19) to find the mean:
\[ \text{Mean} = \frac{650}{19} \approx 34.21 \]
This mean value gives us an idea of the average caffeine content in a can for the sampled brands.
median determination
The median is the middle value in an ordered data set and is useful for understanding the central tendency when dealing with skewed distributions. To find the median:
1. Arrange the data in ascending order: 0, 0, 0, 0, 34, 34, 34, 36, 38, 41, 41, 41, 45, 47, 47, 51, 53, 54, 55.
2. Identify the middle value. With 19 values, the median is the 10th value in the ordered list; thus:
\[ \text{Median} = 41 \]
So, 41 is the median caffeine content among the brands sampled.
mode identification
The mode is the value that appears most frequently in a data set. It's particularly useful for categorical data or for understanding the common occurrences in a data set. In the given caffeine amounts:
\[ \text{Data list: 0, 0, 0, 0, 34, 34, 34, 36, 38, 41, 41, 41, 45, 47, 47, 51, 53, 54, 55} \]
0 appears 4 times and 41 appears 3 times. Since 0 appears the most often, it is the mode:
\[ \text{Mode} = 0 \]
midrange calculation
The midrange is found by averaging the minimum and maximum values in a data set. It is another measure of the central tendency. To calculate the midrange:
1. Identify the minimum value (0) and the maximum value (55) in the data set.
2. Average these values:
\[ \text{Midrange} = \frac{0 + 55}{2} = 27.5 \]
So, the midrange for the caffeine content is 27.5 mg. This measure provides a simple estimate of the center of the range of data values.

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Most popular questions from this chapter

Here are four of the Verizon data speeds (Mbps) from Figure 3-1: \(13.5,10.2,21.1,15.1\). Find the mean and median of these four values. Then find the mean and median after including a fifth value of 142 , which is an outlier. (One of the Verizon data speeds is \(14.2 \mathrm{Mbps}\), but 142 is used here as an error resulting from an entry with a missing decimal point.) Compare the two sets of results. How much was the mean affected by the inclusion of the outlier? How much is the median affected by the inclusion of the outlier?

Use the given data to construct a boxplot and identify the 5-number summary. Fourteen different second-year medical students at Bellevue Hospital measured the blood pressure of the same person. The systolic readings \((\mathrm{mm} \mathrm{Hg})\) are listed below. \(\begin{array}{lllllllllllllll}138 & 130 & 135 & 140 & 120 & 125 & 120 & 130 & 130 & 144 & 143 & 140 & 130 & 150\end{array}\)

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below are the jersey numbers of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII. What do the results tell us? \(\begin{array}{llllllllllll}89 & 91 & 55 & 7 & 20 & 99 & 25 & 81 & 19 & 82 & 60\end{array}\)

Use the given data to construct a boxplot and identify the 5-number summary. Listed below are the measured radiation absorption rates (in \(\mathrm{W} / \mathrm{kg}\) ) corresponding to these cell phones: iPhone 5S, BlackBerry Z30, Sanyo Vero, Optimus V, Droid Razr, Nokia N97, Samsung Vibrant, Sony Z750a, Kyocera Kona, LG G2, and Virgin Mobile Supreme. The data are from the Federal Communications Commission. \(\begin{array}{lllllllllll}1.18 & 1.41 & 1.49 & 1.04 & 1.45 & 0.74 & 0.89 & 1.42 & 1.45 & 0.51 & 1.38\end{array}\)

USA Today published a list consisting of the state tax on each gallon of gas. If we add the 50 state tax amounts and then divide by 50 , we get \(27.3\) cents. Is the value of \(27.3\) cents the mean amount of state sales tax paid by all U.S. drivers? Why or why not?
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