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Chapter 1: Problem 27

Each year, U.S. News and World Report publishes a ranking of U.S. business schools. The following data give the acceptance rates (percentage of applicants admitted) for the best 25 programs in the most recent survey: $$ \begin{array}{lllllllll} 16.3 & 12.0 & 25.1 & 20.3 & 31.9 & 20.7 & 30.1 & 19.5 & 36.2 \\ 46.9 & 25.8 & 36.7 & 33.8 & 24.2 & 21.5 & 35.1 & 37.6 & 23.9 \\ 17.0 & 38.4 & 31.2 & 43.8 & 28.9 & 31.4 & 48.9 & & \end{array} $$ Construct a dotplot, and comment on the interesting features of the plot.

Short Answer

Expert verified
By creating dotplot, you can visualize acceptance rates of top-rated business schools. The dotplot would reveal patterns, outliers and data concentration.

Step by step solution

01

Sorting the data

First, you should sort the data in ascending order. Given numbers can be sorted using either hand-written method or a computer algorithm if the number set is large.
02

Creating a dotplot

After sorting the data, you can draw your dotplot. On a sheet of paper or digital editor, write a horizontal line which represents the range of your data (from smallest to largest). Above the line, you can mark each data point with a dot. More dots stack vertically if certain data points repeat multiple times.
03

Analyzing your dotplot

Now it's time to analyze the plot. Look for patterns and interesting features such as clusters of data points, gaps in the data, and any potential outliers. Note that, outliers are points that are significantly different from most others.

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Key Concepts

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

Acceptance Rates
Acceptance rates show the percentage of applicants who are admitted to a particular program or institution. In education, particularly when it comes to prestigious programs like U.S. business schools, the acceptance rate is a key feature that potential applicants often consider. Acceptance rates can tell us:
  • How competitive a program is - lower rates usually mean higher competition.
  • The overall capacity and selectivity of the program.
For example, if a program has an acceptance rate of 12%, it means that only 12 out of every 100 applicants are admitted. This can imply a highly selective process, indicative of a program striving to maintain a certain level of quality among its entrants. When interpreting acceptance rates:
  • Consider the size of the applicant pool - huge numbers can skew the perception of being selective.
  • Look at trends over time - a sudden drop or rise might indicate significant changes in admission policies.
Understanding these aspects can aid in evaluating an institution's standards and predicting potential outcomes for applicants.
Data Visualization
Data visualization is a critical process in interpreting and understanding large sets of data like acceptance rates. Presenting data in a visual format, such as a dotplot, makes it easier to grasp the information quickly. Some advantages of data visualization:
  • Complex data is simplified, helping in making informed decisions at a glance.
  • Trends, clusters, and anomalies become more apparent.
Dotplots specifically are handy when showing frequency distributions of small data sets. They allow for easy comparison of data points as each point is represented clearly with its stack of dots. To construct a dotplot:
  • Sort your data in ascending order.
  • Draw a horizontal line to represent your range.
  • Plot a dot above the number line for each occurrence of a data point.
With this tool, data patterns stand out vividly, enabling quick recognition of trends and outliers, which are essential for analysis.
Outliers Analysis
Outliers in data are points that lie significantly beyond the general range of the data set. Recognizing outliers is essential because they can indicate anomalies or errors in data collection, or uncover unique insights. When analyzing outliers:
  • Determine if they are true errors or legitimate peculiarities.
  • Consider the context of the data 鈥� sometimes, what seems like an outlier might not be significant depending on the situation.
In the context of acceptance rates, an outlier might be a school with an unusually high or low acceptance rate compared to peers. Such schools might operate under different circumstances or policies, affecting their rates. Identifying outliers on a dotplot involves looking for dots that stand far from the general cluster of data points. Their presence can:
  • Alter the mean and standard deviation considerably, affecting overall analysis.
  • Offer insights into exceptional cases or changes in data trends.
Correctly interpreting outliers helps enhance the overall understanding of the dataset, leading to more accurate conclusions and decisions.

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Most popular questions from this chapter

Nonresponse is a common problem facing researchers who rely on mail questionnaires. In the paper "Reasons for Nonresponse on the Physicians' Practice Survey" ( 1980 Proceedings of the Section on Social Statistics \([1980]: 202), 811\) doctors who did not respond to the AMA Survey of Physicians were contacted about the reason for their nonparticipation. The results are summarized in the accompanying relative frequency distribution. Draw the corresponding bar chart. $$ \begin{array}{lc} \text { Reason } & \begin{array}{l} \text { Relative } \\ \text { Frequency } \end{array} \\ \hline \text { 1. No time to participate } & .264 \\ \text { 2. Not interested } & .300 \\ \text { 3. Don't like surveys in general } & .145 \\ \text { 4. Don't like this particular survey } & .025 \\ \text { 5. Hostility toward the government } & .054 \\ \text { 6. Desire to protect privacy } & .056 \\ \text { 7. Other reason for refusal } & .053 \\ \text { 8. No reason given } & .103 \\ \hline \end{array} $$

"Ozzie and Harriet Don't Live Here Anymore" (San Luis Obispo Tribune, February 26,2002 ) is the title of an article that looked at the changing makeup of America's suburbs. The article states that nonfamily households (e.g. homes headed by a single professional or an elderly widow) now outnumber married couples with children in suburbs of the nation's largest metropolitan areas. The article goes on to state: In the nation's 102 largest metropolitan areas, "nonfamilies" comprised 29 percent of households in 2000 , up from 27 percent in 1990 . While the number of married-with-children homes grew too, the share did not keep pace. It declined from 28 percent to 27 percent. Married couples without children at home live in another 29 percent of suburban households. The remaining 15 percent are single-parent homes. Use the given information on type of household in 2000 to construct a frequency distribution and a bar chart. (Be careful to extract the 2000 percentages from the given information).

Many adolescent boys aspire to be professional athletes. The paper "Why Adolescent Boys Dream of Becoming Professional Athletes" (Psychological Reports [1999]: 1075-1085) examined some of the reasons. Each boy in a sample of teenage boys was asked the following question: "Previous studies have shown that more teenage boys say that they are considering becoming professional athletes than any other occupation. In your opinion, why do these boys want to become professional athletes?" The resulting data are shown in the following table: $$ \begin{array}{lc} \text { Response } & \text { Frequency } \\ \hline \text { Fame and celebrity } & 94 \\ \text { Money } & 56 \\ \text { Attract women } & 29 \end{array} $$ $$ \begin{array}{lc} \text { Response } & \text { Frequency } \\ \hline \text { Like sports } & 27 \\ \text { Easy life } & 24 \\ \text { Don't need an education } & 19 \\ \text { Other } & 19 \\ & \\ \hline \end{array} $$ Construct a bar chart to display these data.

The student senate at a university with 15,000 students is interested in the proportion of students who favor a change in the grading system to allow for plus and minus grades (e.g., \(\mathrm{B}+, \mathrm{B}, \mathrm{B}-\), rather than just \(\mathrm{B}\) ). Two hundred students are interviewed to determine their attitude toward this proposed change. What is the population of interest? What group of students constitutes the sample in this problem?

Data from a poll conducted by Travelocity led to the following estimates: Approximately \(40 \%\) of travelers check work email while on vacation, about \(33 \%\) take cell phones on vacation in order to stay connected with work, and about \(25 \%\) bring a laptop computer on vacation (San Luis Obispo Tribune, December 1, 2005). Are the given percentages population values or were they computed from a sample?
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