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Chapter 3: Problem 27

A random sample of 25 college students was asked, "How many hours per week typically do you work outside the home?" Their responses were as follows: $$ \begin{array}{rrrrr} \hline 0 & 0 & 15 & 20 & 30 \\ \hline 40 & 30 & 20 & 35 & 35 \\ \hline 28 & 15 & 20 & 25 & 25 \\ \hline 30 & 5 & 0 & 30 & 24 \\ \hline 28 & 30 & 35 & 15 & 15 \\ \hline \end{array} $$ Determine the shape of the distribution of hours worked by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes hours worked?

Short Answer

Expert verified
Organize the data. Make a frequency table and histogram. Calculate the mean (21) and median (25). The median better describes hours worked.

Step by step solution

01

Organize the Data

First, list all the given responses in ascending order. This helps in understanding the distribution and calculating the median: 0, 0, 0, 5, 15, 15, 15, 15, 20, 20, 20, 24, 25, 25, 28, 28, 30, 30, 30, 30, 30, 35, 35, 35, 40.
02

Create Frequency Distribution

Next, create a frequency distribution table to count how often each value appears:0: 35: 115: 420: 324: 125: 228: 230: 535: 340: 1.
03

Draw the Histogram

Draw a frequency histogram using the frequency distribution table. The x-axis represents the number of hours worked, and the y-axis represents the frequency. Each bar height should match the frequency of each number of hours worked.
04

Calculate the Mean

To calculate the mean, sum all the responses and divide by the number of responses. Sum: 0+0+0+5+15+15+15+15+20+20+20+24+25+25+28+28+30+30+30+30+30+35+35+35+40 = 525Mean: \(\frac{525}{25} = 21\)
05

Calculate the Median

The median is the middle value in an ordered list. Since there are 25 responses (an odd number), the median is the 13th value:Ordered list: 0, 0, 0, 5, 15, 15, 15, 15, 20, 20, 20, 24, **25**, 25, 28, 28, 30, 30, 30, 30, 30, 35, 35, 35, 40Median: 25
06

Determine the Best Measure of Central Tendency

To determine which measure (mean or median) best describes the data, analyze the distribution shape. The presence of multiple 0s and high values (like 40) suggests potential skewness. In a skewed distribution, the median is usually a better measure because it is less affected by extreme values.In this case, there are outliers (0 hours and 40 hours), thus the median (25) might better describe the typical number of hours worked compared to the mean (21).

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Key Concepts

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

frequency histogram
A frequency histogram provides a visual representation of data distribution. It involves grouping data points into intervals and representing the frequency of each interval with bars. For our exercise, the histogram showcases how many students work a certain number of hours per week. Start with an x-axis representing hours worked and a y-axis for frequency. Each bar's height shows how many students fall into each hour category. For our dataset, you'll see higher bars for multiples of 5 and 15, indicating more common responses.
mean calculation
The mean is commonly referred to as the average. It sums up all data points and divides by the total number of points. For instance, in our problem, we add up each student's working hours and divide by 25. This gives us: \(\frac{525}{25} = 21\). The mean offers a quick snapshot of central tendency, helping summarize the data with a single value.
median calculation
The median represents the middle value of an ordered dataset. To find it, list all numbers from least to greatest and pick the central value. For datasets with an odd number of entries, the median is the middle value. Thus, in our example, after ordering the hours worked, we pick the 13th value: **25**. A median is crucial in understanding the center of a dataset, especially when outliers skew the mean.
skewness in distribution
Skewness in distribution tells us how symmetrical or asymmetrical our data is. A distribution is skewed if it leans heavily toward one side. In our exercise, multiple entries of 0 and high values like 40 suggest a skewed distribution. Positive skew means longer tails on the right, while negative skew means longer tails on the left. Observing skewness helps decide the best measure of central tendency. In skewed distributions, the median usually offers a more reliable average since it isn't influenced by outliers as much as the mean.

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Most popular questions from this chapter

According to a Credit Suisse Survey, the mean net worth of all individuals in the United States in 2014 was \(\$ 301,000,\) while the median net worth was \(\$ 45,000 .\) (a) Which measure do you believe better describes the typical individual's net worth? Support your opinion. (b) What shape would you expect the distribution of net worth to have? Why? (c) What do you think causes the disparity in the two measures of central tendency?

Marissa has just completed her second semester in college. She earned a \(\mathrm{B}\) in her five-hour calculus course, an A in her three-hour social work course, an A in her four-hour biology course, and a \(\mathrm{C}\) in her three-hour American literature course. Assuming that an A equals 4 points, a B equals 3 points, and a C equals 2 points, determine Marissa's gradepoint average for the semester.

Explain the advantage of using \(z\) -scores to compare observations from two different data sets.

A sample of 30 registered voters was surveyed in which the respondents were asked, "Do you consider your political views to be conservative, moderate, or liberal?" The results of the survey are shown in the table. $$ \begin{array}{lll} \hline \text { Liberal } & \text { Conservative } & \text { Moderate } \\ \hline \text { Moderate } & \text { Liberal } & \text { Moderate } \\ \hline \text { Liberal } & \text { Moderate } & \text { Conservative } \\ \hline \text { Moderate } & \text { Conservative } & \text { Moderate } \\ \hline \text { Moderate } & \text { Moderate } & \text { Liberal } \\ \hline \text { Liberal } & \text { Moderate } & \text { Liberal } \\ \hline \text { Conservative } & \text { Moderate } & \text { Moderate } \\ \hline \text { Liberal } & \text { Conservative } & \text { Liberal } \\ \hline \text { Liberal } & \text { Conservative } & \text { Liberal } \\ \hline \text { Conservative } & \text { Moderate } & \text { Conservative } \\\ \hline \end{array} $$ (a) Determine the mode political view. (b) Do you think it would be a good idea to rotate the choices conservative, moderate, or liberal in the question? Why?

The data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds. $$ \begin{array}{lll} \hline 2.0 & 1.9 & 1.8 \\ \hline 3.2 & 2.4 & 3.4 \\ \hline 1.6 & 2.7 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 8.4 & 7.2 & 7.6 \\ \hline 7.4 & 6.9 & 9.4 \\ \hline 9.1 & 8.1 & \\ \hline \end{array} $$ (a) Determine the mean and standard deviation of each data set. (b) Based only on the standard deviation, which data set has more spread? (c) What proportion of the observations is within one standard deviation of the mean for each data set? (d) The coefficient of variation, \(C V\), is defined as the ratio of the standard deviation to the mean of a data set, so $$ C V=\frac{\text { standard deviation }}{\text { mean }} $$ The coefficient of variation is unitless and allows for comparison in spread between two data sets by describing the amount of spread per unit mean. After all, larger numbers will likely have a larger standard deviation simply due to the size of the numbers. Compute the coefficient of variation for both data sets. Which data set do you believe has more "spread"? (e) Let's take this idea one step further. The following data represent the height of a random sample of 8 male college students. The data set on the left has their height measured in inches, and the data set on the right has their height measured in centimeters. $$ \begin{array}{lll} \hline 74 & 68 & 71 \\ \hline 66 & 72 & 69 \\ \hline 69 & 71 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 187.96 & 172.72 & 180.34 \\ \hline 167.64 & 182.88 & 175.26 \\ \hline 175.26 & 180.34 & \\ \hline \end{array} $$
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