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Chapter 3: Problem 20

Standard milk chocolate M\&Ms \(^{T M}\) come in six colors. The Fun Size bags typically contain between 16 and 20 candies, so it is common for a Fun Size bag to have some of the six colors missing. Each of the 14 students in a summer statistics class was given a Fun Size bag and asked to count the number of colors present in the bag. The following data are the number of colors found in these 14 bags: \(\begin{array}{llllllllllllll}3 & 6 & 5 & 4 & 6 & 3 & 2 & 5 & 5 & 4 & 5 & 6 & 3 & 4\end{array}\)

Short Answer

Expert verified
The mean number of M&M colors in the bags is approximately 4.29, the modes are 4, 5 and 6, the median is 4.5, and the range is 4.

Step by step solution

01

Arrange the data in order

To simplify further calculations, it is helpful to first organize the data in order of magnitude. So the sorted data would be: 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6.
02

Compute the mean

The mean (average) is found by adding all numbers in the data and dividing by the number of items in the data. In this case, the mean can be computed as: \( \frac{2+3+3+3+4+4+4+5+5+5+5+6+6+6}{14} \approx 4.29 \)
03

Determine the mode

The mode is the value that appears most frequently in a data set. Looking at the sorted data, we can see that the numbers 4, 5 and 6 each appear three times and number 3 appears two times. So, there are three modes in this data: 4, 5 and 6.
04

Calculate the median

The median value is the middle one in a data distribution when the data have been arranged in order of magnitude. In this case, since there are 14 numbers, the median would be the average of the 7th and 8th numbers. Calculating the median gives us: \( \frac{4+5}{2} = 4.5 \)
05

Calculate the range

The range of a dataset is the difference between the highest and the lowest values. In this dataset, the range can be calculated as: \(6 - 2 = 4\)

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Key Concepts

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

Mean Calculation
Mean calculation is a key concept in descriptive statistics that helps us find the average of a data set. It's like finding the middle ground: Imagine you have several numbers and want to know what the collective balance is. To calculate the mean, simply add together all numbers and divide by the total count of numbers.

For instance, let鈥檚 use the sorted M&M data: 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, and 6. Add them up to get 60. Since there are 14 numbers, we divide 60 by 14 to find the mean:

\[\text{Mean} = \frac{60}{14} \approx 4.29\]

This mean value of approximately 4.29 describes the average number of colors in each M&M bag.
Mode Identification
The mode in a data set is the number that appears most frequently. Imagine you are tracking your favorite candy color; whichever one shows up the most is like the superstar of your M&M bag!

For our set, we have 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, and 6. Count the occurrences:
  • 3 appears 3 times
  • 4 appears 3 times
  • 5 appears 4 times
  • 6 appears 3 times
The number 5 appears most frequently, making it the mode. But, numbers 4, 5, and 6 all show up three times, so we actually have three modes: 4, 5, and 6 in this dataset. In statistics, having more than one mode means the data is "multimodal." It's like having multiple popular colors in a bag of candies!
Median Determination
The median is like the midpoint mark, dividing a dataset into two equal halves. To find this, we first sort the data from smallest to largest. Then, based on the count of the numbers, we locate the middle number or average the two middle numbers if needed.

Our sorted data is: 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6. Since we have 14 data points, we'll take the 7th and 8th numbers and average them:

\[\text{Median} = \frac{4+5}{2} = 4.5\]

This tells us that 4.5 is right in the middle of our dataset, dividing the lower half of candy colors from the upper half.
Range Calculation
Finding the range helps us understand how diverse our dataset is. It tells us the spread from the smallest to the largest number, or how much swing there is in your M&M bag's colors.

In our example, we'll take the smallest number, 2, and the largest number, 6, and subtract:

\[\text{Range} = 6 - 2 = 4\]

Here, the range is 4, indicating that there is a spread of 4 colors between the least and most colorful M&M bags. It's a simple measure giving insight into data variability.

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Most popular questions from this chapter

Consider the following two data sets. \(\begin{array}{llrlrl}\text { Data Set I: } & 4 & 8 & 15 & 9 & 11 \\ \text { Data Set II: } & 8 & 16 & 30 & 18 & 22\end{array}\) Note that each value of the second data set is obtained by multiplying the corresponding value of the first data set by 2. Calculate the standard deviation for each of these two data sets using the formula for population data. Comment on the relationship between the two standard deviations.

One disadvantage of the standard deviation as a measure of dispersion is that it is a measure of absolute variability and not of relative variability. Sometimes we may need to compare the variability of two different data sets that have different units of measurement. The coefficient of variation is one such measure. The coefficient of variation, denoted by CV, expresses standard deviation as a percentage of the mean and is computed as follows: For population data: \(\mathrm{CV}=\frac{\sigma}{\mu} \times 100 \%\) For sample data: $$ \mathrm{CV}=\frac{s}{\bar{x}} \times 100 \% $$ The yearly salaries of all employees who work for a company have a mean of \(\$ 62,350\) and a standard deviation of \(\$ 6820\). The years of experience for the same employees have a mean of 15 years and a standard deviation of 2 years. Is the relative variation in the salaries larger or smaller than that in years of experience for these employees?

Melissa's grade in her math class is determined by three 100 -point tests and a 200 -point final exam. To determine the grade for a student in this class, the instructor will add the four scores together and divide this sum by 5 to obtain a percentage. This percentage must be at least 80 for a grade of \(\mathrm{B}\). If Melissa's three test scores are 75,69 , and 87 , what is the minimum score she needs on the final exam to obtain a B grade?

The prices of all college textbooks follow a bell-shaped distribution with a mean of \(\$ 180\) and a standard deviation of \(\$ 30\). A. Using the empirical rule, find the percentage of all college textbooks with their prices between 1\. \(\$ 150\) and \(\$ 210\) ii. \(\$ 120\) and \(\$ 240\) *b. Using the empirical rule, find the interval that contains the prices of \(99.7 \%\) of college textbooks.

The Belmont Stakes is the final race in the annual Triple Crown of thoroughbred horse racing. The race is \(1.5\) miles in length, and the record for the fastest time of 2 minutes, 24 seconds is held by Secretariat, the 1973 winner. We compared Secretariat's time from 1973 with the time of each winner of the Belmont Stakes for the years \(1999-2011\). The following data represent the differences (in seconds) between each winner's time for the years \(1999-2011\) and Secretariat's time in 1973 . For example, the 1999 winner took \(3.80\) seconds longer than Secretariat to finish the race. \(\begin{array}{lllllllllllllll}3.80 & 7.20 & 2.80 & 5.71 & 4.26 & 3.50 & 4.75 & 3.81 & 4.74 & 5.65 & 3.54 & 7.57 & 6.88\end{array}\) a. Calculate the mean and median. Do these data have a mode? Why or why not? b. Compute the range, variance, and standard deviation for these data.
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