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Chapter 1: Problem 37

Students in a college statistics class responded to a survey designed by their teacher. One of the survey questions was "How much sleep did you get last night?" Here are the data (in hours): $$ \begin{array}{lrrrrrrrrrrrr} \hline 9 & 6 & 8 & 6 & 8 & 8 & 6 & 6.5 & 6 & 7 & 9 & 4 & 3 & 4 \\ 5 & 6 & 11 & 6 & 3 & 6 & 6 & 10 & 7 & 8 & 4.5 & 9 & 7 & 7 \\ \hline \end{array} $$ (a) Make a dotplot to display the data. (b) Describe the overall pattern of the distribution and any departures from that pattern.

Short Answer

Expert verified
Create a dotplot by marking each value as dots on a number line. The distribution is centered around 7 hours, slightly right-skewed, with a potential outlier at 11 hours.

Step by step solution

01

Organize the Data

The first step is to organize the given data in a way that makes it easy to plot. Order the data in ascending sequence to better understand the distribution. The ordered data is:\[3, 3, 4, 4, 4.5, 5, 6, 6, 6, 6, 6, 6.5, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 11\]
02

Create the Dotplot

For a dotplot, assign each distinct value in the data set to a point on a number line, and place a dot above the number line for each occurrence of that value. For example, the number 6 appears six times, so there should be six dots above the number 6. This graphical representation will show the frequency of each data point straightforwardly.
03

Analyze the Dotplot

Inspect the dotplot to identify the overall pattern and any unusual features. Observe the central tendency, variability, and shape of the data distribution. Check for any values that are notably higher or lower than the rest, these would be considered outliers.
04

Describe the Distribution

The dotplot reveals that most of the values are clustered around 6 to 8 hours, indicating a center around 7 hours. The data is relatively symmetric but slightly right-skewed due to the presence of a few higher values (10, 11). The variability is moderate, with values ranging from 3 to 11 hours. The point at 11 could be seen as a potential outlier since it is quite far from the other data points.

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Key Concepts

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

Data Distribution
In statistics, understanding data distribution is crucial. It refers to how the values of a dataset are spread or distributed over a range. Analyzing data distribution helps determine where most of the values lie, identifying patterns or anomalies in the data.
Distributions can vary widely between datasets and are typically categorized based on their shape, spread, and center. When examining a dataset like the sleep hours reported by students:
  • The shape of the distribution reveals how data is clustered, whether it's symmetrical, skewed, or forms a specific pattern.
  • The spread pertains to the range of the data, showing the variability and how bunched or spaced the data points are.
  • The center gives insight into a central point around which the data values converge. This often involves measures of central tendency like mean or median.
In our example, the sleep hours data shows a right-skewed distribution due to higher outlying values, creating a spread mainly around 6-8 hours with some reaching as high as 11.
Dotplot
A dotplot is a simple statistical chart used to visualize data distribution. It represents each data point as a dot plotted along a number line. This visual tool is particularly beneficial for small datasets, providing a clear picture of how data points stack up.
To create a dotplot:
  • Start by drawing a number line that spans the range of your data values.
  • Assign each distinct data value its position on the line.
  • Place a dot above the number line for each occurrence of a value. Multiple dots stack vertically when values repeat.
For instance, the number 6 in our dataset appears six times, resulting in six dots stacked above the number 6 on the dotplot. This straightforward representation helps quickly identify modes, trends, and any outliers present in the dataset.
Data Analysis
Data analysis involves examining datasets to draw meaningful conclusions. It's a core skill in statistics education, allowing students to make sense of data arrangements like dotplots and beyond.
When analyzing data:
  • Look for overall patterns, such as whether data tends to cluster around certain values or if modes appear.
  • Identify any gaps or abnormal spikes that could suggest outliers or separate trends.
  • Assess symmetry or skewness to understand the distribution shape and its implications.
For the sleep hours survey, data analysis reveals a concentration of responses around 6-8 hours. Note the skewness caused by responses at 10 and 11 hours. Recognizing these trends helps in predicting behaviors or outcomes in similar future scenarios.
Central Tendency
Central tendency measures are statistical metrics that indicate the center or typical value of a dataset. They provide insight into what constitutes an average or most common data point.
The three most common measures of central tendency include:
  • Mean: The arithmetic average, calculated by summing all values and dividing by the count. For the sleep example, adding all hours then dividing by the total entries gives the mean sleep time.
  • Median: The middle value when data is ordered. If there's an even number of data points, the median is the average of the two middle values. In this dataset, it portrays the center of the sleep duration based on positioning, unaffected by extreme values.
  • Mode: The most frequently occurring value, highlighting commonality. Here, the mode is around the 6-hour mark, showing many students reported similar sleep patterns.
Understanding these concepts aids in summarizing and simplifying complex data into comprehensible information.

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Most popular questions from this chapter

Here are data from a survey conducted at eight high schools on smoking among students and their parents: \({ }^{17}\) $$ \begin{array}{lccc} \hline & \begin{array}{c} \text { Neither } \\ \text { parent } \\ \text { smokes } \end{array} & \begin{array}{c} \text { One } \\ \text { parent } \\ \text { smokes } \end{array} & \begin{array}{c} \text { Both } \\ \text { parents } \\ \text { smoke } \end{array} \\ \text { Student does not smoke } & 1168 & 1823 & 1380 \\ \text { Student smokes } & 188 & 416 & 400 \\ \hline \end{array} $$ (a) How many students are described in the two-way table? What percent of these students smoke? (b) Give the marginal distribution (in percents) of parents" smoking behavior, both in counts and in percents.

There are many ways to measure the reading ability of children. One frequently used test is the Degree of Reading Power (DRP). In a research study on third- grade students, the DRP was administered to 44 students. \({ }^{32}\) Their scores were: $$ \begin{array}{lllllllllll} \hline 40 & 26 & 39 & 14 & 42 & 18 & 25 & 43 & 46 & 27 & 19 \\ 47 & 19 & 26 & 35 & 34 & 15 & 44 & 40 & 38 & 31 & 46 \\ 52 & 25 & 35 & 35 & 33 & 29 & 34 & 41 & 49 & 28 & 52 \\ 47 & 35 & 48 & 22 & 33 & 41 & 51 & 27 & 14 & 54 & 45 \\ \hline \end{array} $$ Make a histogram to display the data. Write a paragraph describing the distribution of DRP scores.

Last year a small accounting firm paid each of its five clerks \(\$ 22,000\), two junior accountants \(\$ 50,000\) each, and the firm's owner \(\$ 270,000 .\) What is the mean salary paid at this firm? How many of the employees earn less than the mean? What is the median salary? Write a sentence to describe how an unethical recruiter could use statistics to mislead prospective employees.

How long do people travel each day to get to work? The following table gives the average travel times to work (in minutes) for workers in each state and the District of Columbia who are at least 16 years old and don't work at home. \({ }^{30}\) (a) Make a histogram of the travel times using classes of width 2 minutes, starting at 14 minutes. That is, the first class is 14 to 16 minutes, the second is 16 to 18 minutes, and so on. (b) The shape of the distribution is a bit irregular. Is it closer to symmetric or skewed? Describe the center and spread of the distribution. Are there any outliers?

The U.S. Food and Drug Administration (USFDA) limits the amount of caffeine in a 12 -ounce can of carbonated beverage to 72 milligrams. That translates to a maximum of 48 milligrams of caffeine per 8 -ounce serving. Data on the caffeine content of popular soft drinks (in milligrams per 8-ounce serving) are displayed in the stemplot below. $$ \begin{array}{l|l} 1 & 556 \\ 2 & 033344 \\ 2 & 55667778888899 \\ 3 & 113 \\ 3 & 55567778 \\ 4 & 33 \\ 4 & 77 \end{array} $$ (a) Why did we split stems? (b) Give an appropriate key for this graph. (c) Describe the shape, center, and spread of the distribution. Are there any outliers?
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