91影视

In a study of author productivity ("Lotka's Test," Collection Manage., 1982: 111-118), a large number of authors were classified according to the number of articles they had published during a certain period. The results were presented in the accompanying frequency distribution: \(\begin{array}{lrrrrrrrrr}\begin{array}{l}\text { Number of } \\ \text { papers }\end{array} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & \\ \text { Frequency } & 784 & 204 & 127 & 50 & 33 & 28 & 19 & 19 & \\ \text { Number of } & & & & & & & & & \\ \text { papers } & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\\ \text { Frequency } & 6 & 7 & 6 & 7 & 4 & 4 & 5 & 3 & 3\end{array}\) a. Construct a histogram corresponding to this frequency distribution. What is the most interesting feature of the shape of the distribution? b. What proportion of these authors published at least five papers? At least ten papers? More than ten papers? c. Suppose the five 15 's, three 16 's, and three 17 's had been lumped into a single category displayed as \(" \geq 15 . "\) Would you be able to draw a histogram? Explain. d. Suppose that instead of the values 15,16 , and 17 being listed separately, they had been combined into a \(15-17\) category with frequency 11. Would you be able to draw a histogram? Explain.

Step by step solution

01

Construct the Histogram

To construct the histogram, plot the number of papers on the x-axis and the frequency on the y-axis. Each bar represents the frequency of authors who published a certain number of papers. For example, for 1 paper, draw a bar up to 784 on the y-axis. Do this for each number of papers (from 1 to 17). This graphical representation will help visualize the distribution of the data.

02

Analyze Histogram Shape

With the histogram drawn, observe its shape. The distribution is right-skewed, as most authors published fewer papers, and fewer authors published more papers. The tail of the histogram extends to the right, highlighting this skew.

03

Calculate Proportion for At Least Five Papers

Find the sum of frequencies for authors who published at least five papers (5 and above). Calculate this as \( 33 + 28 + 19 + 19 + 6 + 7 + 6 + 7 + 4 + 4 + 5 + 3 + 3 = 144 \). Divide by the total number of authors, which is the sum of all frequencies, \( 1287 \). The proportion is approximately \( \frac{144}{1287} \approx 0.112 \).

04

Calculate Proportion for At Least Ten Papers

Add the frequencies for authors who published at least ten papers (10 and above): \( 7 + 6 + 7 + 4 + 4 + 5 + 3 + 3 = 39 \). The proportion is \( \frac{39}{1287} \approx 0.030 \).

05

Calculate Proportion for More Than Ten Papers

Add the frequencies for authors who published more than ten papers: \( 6 + 7 + 4 + 4 + 5 + 3 + 3 = 32 \). The proportion is \( \frac{32}{1287} \approx 0.025 \).

06

Evaluate Histogram with Combined Category 鈥溾墺15鈥�

If categories 15, 16, and 17 are combined into 鈥溾墺15鈥�, it affects the continuity of numerical categories, thus limiting the histogram鈥檚 clarity on paper distinction. This results in a less precise histogram since this single category does not represent a range uniformly.

07

Evaluate Histogram with 15-17 Category

Combining 15, 16, and 17 into a 15-17 category with a frequency of 11 maintains categories with equal width, allowing each frequency to be approximately represented. This continuity allows for a histogram, albeit with less specific detail for those categories.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Frequency Distribution

A frequency distribution is a way to organize data to show how often things happen. This helps people see what is going on with their data. Imagine you are counting how many books different authors wrote. Each number of books represents one category. How many authors wrote just one book? How many wrote two? By listing these and their counts, you form a frequency distribution. This is especially helpful with large sets of data. It neatly displays data and lets you analyze the patterns. For instance, in our example, authors' productivity was organized by the number of papers they published. This gave us a clear picture of how productive different authors are over time.

Right-Skewed Distribution

A right-skewed distribution is where the bulk of the data points are on the left side, while the tail extends to the right. This means that most of the values are lower, but there are a few values that are much higher. In terms of our histogram, more authors have fewer publications, creating a longer "tail" on the right side. Here are the key points:

• More authors published fewer papers.
• Distribution is not symmetrical.
• The tail reflects the lesser number of authors with many papers.

Understanding this helps in predicting outcomes and assessing variation in data.

Proportion Calculation

Proportion calculation is figuring out how big one part is relative to a whole. It鈥檚 very useful in situations like determining how many authors publish a certain number of papers. To find this proportion:

• Add the number of authors in your category of interest.
• Divide this by the total number of authors.
• Multiply by 100 to get a percentage if desired.

In our exercise, finding the proportion of authors who published at least five papers involved adding frequencies for these categories and dividing by the total number of authors. This calculation helps to understand how common or rare certain outcomes are relative to the entire group.

Data Visualization

Data visualization transforms numbers into graphs or charts, making patterns easier to understand. Histograms are a common method for visualizing frequency distributions. They use bars to represent data, where each bar shows the frequency of a specific category. The length of each bar corresponds to the count of items in each category. Consider when plotting the histogram for author productivity:

• Each bar corresponds to how many authors published a specific number of papers.
• Observe the shape to recognize data distribution (e.g., right-skewed).
• Histograms allow easy comparison between different categories.

Clearly, this visualization lets viewers quickly interpret data trends without diving deep into raw numbers, facilitating better decision-making and insights.