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Chapter 3: Problem 10

The number of foreign workers' certificates for the New England states and the northwestern states is shown. Find the mean, median, and mode for both areas and compare the results. $$\begin{array}{cc}\text { New England states } & \text { Northwestern states } \\\\\hline 6768 & 1870 \\\3196 & 622 \\\1112 & 620 \\\819 & 23 \\\1019 & 172 \\\ 1795 & 112\end{array}$$

Short Answer

Expert verified
New England has higher mean and median; neither region has a mode.

Step by step solution

01

Organize and Sort the Data

First, write down the given values for each region separately and sort the numbers in ascending order. New England states: 819, 1019, 1112, 1795, 3196, 6768 Northwestern states: 23, 112, 172, 620, 622, 1870.
02

Calculate the Mean

To find the mean, add up all the values and then divide by the number of values.For New England states:\[\text{Mean} = \frac{819 + 1019 + 1112 + 1795 + 3196 + 6768}{6}= \frac{14709}{6} = 2451.5\]For Northwestern states:\[\text{Mean} = \frac{23 + 112 + 172 + 620 + 622 + 1870}{6}= \frac{3419}{6} = 569.83\]
03

Calculate the Median

The median is the middle value, or the average of the two middle values if there is an even number of entries.For New England states:\[\text{Median} = \frac{1112 + 1795}{2} = \frac{2907}{2} = 1453.5\]For Northwestern states:\[\text{Median} = \frac{172 + 620}{2} = \frac{792}{2} = 396\]
04

Determine the Mode

The mode is the value that appears most frequently. For New England states, there is no mode since all values are unique. For Northwestern states, there is no mode since all values are unique.
05

Compare the Results

Compare the mean, median, and mode for the two regions. For means, New England has a higher mean (2451.5) compared to Northwestern (569.83). For medians, New England's median (1453.5) is higher than Northwestern's (396). There is no mode for either region since no number repeats.

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Key Concepts

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

Mean
The mean, often referred to as the "average," is a foundational concept in descriptive statistics. It provides a simple summary of a data set by calculating the central value. To find the mean, you start by adding all values in the dataset together. Once you have the total sum, divide this by the count of the numbers in the dataset.
For example, in calculating the mean number of foreign workers' certificates for the New England states, we add the numbers: 819, 1019, 1112, 1795, 3196, and 6768. This sums up to 14709. Since there are six numbers, you divide this total by 6 resulting in a mean value of 2451.5. The mean is an important statistic because it offers a sense of balance in the data. However, it can be influenced by extremely high or low values, such as 6768 in the New England states' dataset, which can skew the mean upwards.
Median
The median offers another perspective on the central tendency of a data set, distinct from the mean. It represents the middle point of a dataset when all values are arranged in ascending order. If the count of numbers is odd, the median is the single middle number. If even, it is the average of the two middle numbers. In the case of the New England states' dataset: 819, 1019, 1112, 1795, 3196, and 6768, you notice there are six numbers. Take the third and fourth numbers (1112 and 1795), average them, and find a median of 1453.5.
This shows the "middle" workers' certificate issue rate. For the Northwestern states, the calculation follows similarly, highlighting a median of 396 when placing numbers in order. The median is a robust measure of central tendency, especially useful in datasets with outliers, as it is not swayed by extremely high or low values.
Mode
In descriptive statistics, the mode refers to the value that appears most frequently in a dataset. Determining the mode is particularly straightforward because it involves observing frequency rather than computing a mathematical point. In both the New England and Northwestern datasets provided, each number appears only once. This means that there is no mode for either set because no number is repeated frequently. Although absent in these datasets, modes can be crucial when analyzing data that includes many repeated values. They offer insights into most common occurrences, such as the most common age group in a population or the most chosen product in a sales report.

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Most popular questions from this chapter

Which is a better relative position, a score of 83 on a geography test that has a mean of 72 and a standard deviation of \(6,\) or a score of 61 on an accounting test that has a mean of 55 and a standard deviation of \(3.5 ?\)

The geometric mean (GM) is defined as the \(n\) th root of the product of \(n\) values. The formula is $$\mathrm{GM}=\sqrt[n]{\left(X_{1}\right)\left(X_{2}\right)\left(X_{3}\right) \cdots\left(X_{n}\right)}$$ The geometric mean of 4 and 16 is $$\mathrm{GM}=\sqrt{(4)(16)}=\sqrt{64}=8$$ The geometric mean of \(1,3,\) and 9 is $$\mathrm{GM}=\sqrt[3]{(1)(3)(9)}=\sqrt[3]{27}=3$$ The geometric mean is useful in finding the average of percentages, ratios, indexes, or growth rates. For example, if a person receives a \(20 \%\) raise after 1 year of service and a \(10 \%\) raise after the second year of service, the average percentage raise per year is not 15 but \(14.89 \%,\) as shown. $$\mathrm{GM}=\sqrt{(1.2)(1.1)} \approx 1.1489$$ Or $$\mathrm{GM}=\sqrt{(120)(110)} \approx 114.89 \%$$ His salary is \(120 \%\) at the end of the first year and \(110 \%\) at the end of the second year. This is equivalent to an average of \(14.89 \%\), since \(114.89 \%-100 \%=\) \(14.89 \% .\) This answer can also be shown by assuming that the person makes \(\$ 10,000\) to start and receives two raises of \(20 \%\) and \(10 \%\). $$\begin{array}{l}\text { Raise } 1=10,000 \cdot 20 \%=\$ 2000 \\\\\text { Raise } 2=12,000 \cdot 10 \%=\$ 1200\end{array}$$ Find the geometric mean of each of these. a. The growth rates of the Living Life Insurance Corporation for the past 3 years were \(35,24,\) and \(18 \%\). b. A person received these percentage raises in salary over a 4-year period: \(8,6,4,\) and \(5 \%\). c. A stock increased each year for 5 years at these percentages: \(10,8,12,9,\) and \(3 \%\). d. The price increases, in percentages, for the cost of food in a specific geographic region for the past 3 years were \(1,3,\) and \(5.5 \% .\)

The average full time faculty member in a post secondary degree-granting institution works an average of 53 hours per week. a. If we assume the standard deviation is 2.8 hours, what percentage of faculty members work more than 58.6 hours a week? b. If we assume a bell-shaped distribution, what percentage of faculty members work more than 58.6 hours a week?

Pearson Coefficient of Skewness A measure to determine the skewness of a distribution is called the Pearson coefficient \((P C)\) of skewness. The formula is $$\mathrm{PC}=\frac{3(\bar{X}-\mathrm{MD})}{s}$$ The values of the coefficient usually range from -3 to +3 . When the distribution is symmetric, the coefficient is zero; when the distribution is positively skewed, it is positive; and when the distribution is negatively skewed, it is negative. Using the formula, find the coefficient of skewness for each distribution, and describe the shape of the distribution. a. Mean \(=10,\) median \(=8,\) standard deviation \(=3\) b. Mean \(=42,\) median \(=45,\) standard deviation \(=4\). c. Mean \(=18.6,\) median \(=18.6,\) standard deviation \(=1.5 .\) d. Mean \(=98,\) median \(=97.6,\) standard deviation \(=4\).

The data show the number of children U.S. Presidents through Obama, had. Construct an ungrouped frequency distribution and find the mean and modal class. $$\begin{array}{llllllllll}0 & 5 & 6 & 0 & 3 & 4 & 1 & 4 & 10 & 8 \\\7 & 0 & 6 & 1 & 0 & 3 & 4 & 5 & 4 & 8 \\\7 & 3 & 5 & 2 & 1 & 2 & 1 & 5 & 3 & 3 \\\0 & 0 & 2 & 2 & 6 & 1 & 2 & 3 & 2 & 2 \\\4 & 4 & 2 & 6 & 1 & 2 & 2 & & &\end{array}$$
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