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Chapter 3: Problem 16

The following data give the numbers of car thefts that occurred in a city during the past 12 days. \(\begin{array}{lccccccccccc}6 & 3 & 7 & 11 & 4 & 3 & 8 & 7 & 2 & 6 & 9 & 15\end{array}\) Find the mean, median, and mode.

Short Answer

Expert verified
The mean is 6.75, the median is 6.5, and the modes are 3, 6, and 7.

Step by step solution

01

Calculate Mean

To calculate the mean, we need to add all the numbers together and then divide the sum by the number of items in the set. So, the mean is calculated as \(\frac{(6+3+7+11+4+3+8+7+2+6+9+15)}{12}\).
02

Calculate Median

The median requires arranging the data in ascending order: 2, 3, 3, 4, 6, 6, 7, 7, 8, 9, 11, 15. As there are 12 numbers in the dataset and hence it's even, the median is the average of the two middle numbers, which are 6 and 7. So Median = \(\frac{(6+7)}{2}\).
03

Calculate Mode

The mode is the number that appears most frequently. In the data set, the numbers 3, 6, and 7 each appear twice. So, the dataset has three modes which are 3, 6, and 7.

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Key Concepts

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

Mean Calculation
In statistics, the mean is a measure that represents the average of a data set. It gives us an idea of the central point of a group of numbers. Calculating the mean involves a straightforward process. First, sum up all the numbers in the data set. Then, divide this sum by the total count of numbers.
For our given data \[6, 3, 7, 11, 4, 3, 8, 7, 2, 6, 9, 15\]we start by adding these numbers together:
  • Sum = 6 + 3 + 7 + 11 + 4 + 3 + 8 + 7 + 2 + 6 + 9 + 15
Once added, divide the total by 12, since there are 12 numbers:
  • Mean = \( \frac{81}{12} \) = 6.75
In conclusion, the mean number of car thefts is 6.75 over the period of 12 days.
Median Determination
The median offers a measure of the center that is less affected by extreme values compared to the mean. To determine the median, we need to arrange the numbers in ascending order. With our data:
  • Ordered Data: 2, 3, 3, 4, 6, 6, 7, 7, 8, 9, 11, 15
Since the number of observations is even (12 observations), the median is the average of the two middle numbers. In this dataset, these numbers are 6 and 7. Therefore, calculate as follows:
  • Median = \( \frac{6 + 7}{2} \)
This equates to \(6.5\).
As such, the median value, or the central tendency, for car thefts within these 12 days is 6.5.
Mode Identification
The mode of a data set is the number that occurs most frequently. Unlike mean and median, a data set can have more than one mode or none at all.
To identify the mode in our data, we count how frequently each number appears:
  • 3 appears 2 times
  • 6 appears 2 times
  • 7 appears 2 times
The numbers 3, 6, and 7 all appear twice, becoming the modes of this data set.
This situation where multiple numbers appear with the same maximum frequency, e.g., bimodal or trimodal, is common in datasets.
Thus, it is essential to note that in this scenario, the dataset has three modes: 3, 6, and 7. This indicates there were similar frequencies in car theft incidents on those days.

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Most popular questions from this chapter

Each year the faculty at Metro Business College chooses 10 members from the current graduating class that they feel are most likely to succeed. The data below give the current annual incomes (in thousands of dollars) of the 10 members of the class of 2000 who were voted most likely to succeed. \(\begin{array}{llllllllll}59 & 68 & 84 & 78 & 107 & 382 & 56 & 74 & 97 & 60\end{array}\) a. Calculate the mean and median. b. Does this data set contain any outlier(s)? If yes, drop the outlier(s) and recalculate the mean and median. Which of these measures changes by a greater amount when you drop the outlier(s)? c. Is the mean or the median a better summary measure for these data? Explain.

The following data give the time (in minutes) that each of 20 students selected from a university waited in line at their bookstore to pay for their textbooks in the beginning of the Fall 2009 semester. \(\begin{array}{rrrrrrrrrr}15 & 8 & 23 & 21 & 5 & 17 & 31 & 22 & 34 & 6 \\ 5 & 10 & 14 & 17 & 16 & 25 & 30 & 3 & 31 & 19\end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.

Briefly describe how the three quartiles are calculated for a data set. Illustrate by calculating the three quartiles for two examples, the first with an odd number of observations and the second with an even number of observations.

Suppose that on a certain section of I-95 with a posted speed limit of \(65 \mathrm{mph}\), the speeds of all vehicles have a bell-shaped distribution with a mean of \(72 \mathrm{mph}\) and a standard deviation of \(3 \mathrm{mph}\). a. Using the empirical rule, find the percentage of vehicles with the following speeds on this section of I-95. i. 63 to \(81 \mathrm{mph}\) ii. 69 to \(75 \mathrm{mph}\) *b. Using the empirical rule, find the interval that contains the speeds of \(95 \%\) of vehicles traveling on this section of \(\mathrm{I}-95\).

Prepare a box-and-whisker plot for the following data: \(\begin{array}{llllllll}36 & 43 & 28 & 52 & 41 & 59 & 47 & 61 \\ 24 & 55 & 63 & 73 & 32 & 25 & 35 & 49 \\ 31 & 22 & 61 & 42 & 58 & 65 & 98 & 34\end{array}\)
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