91Ó°ÊÓ

In a November 2008 Harris Poll, U.S. adults were asked "Will the Obama administration be too liberal or conservative?" Of the respondents, \(35 \%\) said that it will be too liberal (L), \(43 \%\) said that it will be neither too liberal nor too conservative (N), \(4 \%\) said that it will be too conservative (C), and \(18 \%\) said that they do not know (K). In a recent poll, 40 people were asked whether the Obama administration has been too liberal or too conservative. Their responses are given below. $$ \begin{array}{llllllllll} \mathrm{L} & \mathrm{N} & \mathrm{K} & \mathrm{K} & \mathrm{C} & \mathrm{L} & \mathrm{K} & \mathrm{K} & \mathrm{L} & \mathrm{L} \\ \mathrm{K} & \mathrm{K} & \mathrm{L} & \mathrm{K} & \mathrm{N} & \mathrm{N} & \mathrm{N} & \mathrm{K} & \mathrm{N} & \mathrm{K} \\ \mathrm{N} & \mathrm{K} & \mathrm{K} & \mathrm{N} & \mathrm{L} & \mathrm{L} & \mathrm{N} & \mathrm{N} & \mathrm{K} & \mathrm{K} \\ \mathrm{L} & \mathrm{K} & \mathrm{L} & \mathrm{N} & \mathrm{L} & \mathrm{L} & \mathrm{N} & \mathrm{N} & \mathrm{K} & \mathrm{K} \end{array} $$ a. Prepare a frequency distribution for these data. b. Calculate the relative frequencies and percentages for all classes. c. Draw a bar graph for the frequency distribution and a pie chart for the percentage distribution. d. What percentage of these respondents said "too liberal"?

Step by step solution

01

Break down the data

Count the number of times each response appears in the data. Check for 'L', 'N', 'K', and 'C'.

02

Prepare frequency distribution

Group the data by different types of responses ('L', 'N', 'K', 'C') and count the number of occurrences. Create a table to represent these data. The table should include columns for responses and respective frequencies.

03

Calculate relative frequencies and percentages

Divide each frequency by the total number of respondents to get the relative frequency. Multiply the relative frequency by 100 to convert it into a percentage. Add this information to the frequency distribution table.

04

Draw a bar chart and pie chart

Use the frequency distribution to draw a bar graph. In this graph, the 'x-axis' represents the types of responses while the 'y-axis' represents the number of respondents. All bars are of equal width and the height of each represents its corresponding frequency. Draw a pie chart using the percentages. Divide the pie into segments that correspond to each response type in proportion to the percentages.

05

Find percentage of 'too liberal' respondents

Look at the frequency distribution table and identify the percentage corresponding to the 'L' (too liberal) responses.

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Key Concepts

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

Frequency Distribution

Frequency distribution is a way to organize data to show how often each different response occurs. Imagine counting how many people gave a particular answer in a survey. This is exactly what frequency distribution does. It's like sorting the responses into different boxes and seeing how many pieces—or opinions—are in each box. This helps us quickly understand the data without getting lost in a sea of numbers.
In our example, the survey had four possible responses—'L', 'N', 'C', and 'K'. We counted each occurrence:

• 'L' stands for "Too Liberal" and occurred 10 times.
• 'N' stands for "Neither too liberal nor too conservative" and appeared 12 times.
• 'C' stands for "Too Conservative" and appeared only once.
• 'K' stands for "Do Not Know" and occurred 17 times.

This count of responses is the frequency distribution. It's a simple yet powerful tool that lays the foundation for deeper analysis.

Relative Frequency

Relative frequency gives a better picture by showing how each type of response relates to the total number of responses. Instead of raw counts, it tells us the proportion of each response in the whole dataset. Think of it as understanding how big each "slice of the pie" is.
To calculate relative frequency, take the frequency of each response and divide it by the total number of responses (which is 40 in our survey).

• For 'L', the relative frequency is \( rac{10}{40} = 0.25 \).
• For 'N', it's \( rac{12}{40} = 0.30 \).
• For 'C', it's \( rac{1}{40} = 0.025 \).
• For 'K', it's \( rac{17}{40} = 0.425 \).

Relative frequencies are often multiplied by 100 to turn them into percentages, making it easier to understand. For example, the relative frequency of 0.25 becomes 25%.

Bar Graph

Bar graphs offer a visual interpretation of the frequency distribution, making it easy to see which responses are most common. Each bar in the graph represents a different response type and the height of each bar shows how many times that response was given.
Here is how you can interpret a bar graph:

• The x-axis (horizontal) lists the types of responses: 'L', 'N', 'C', and 'K'.
• The y-axis (vertical) shows the number of responses for each type.
• The taller the bar, the more frequently that response was given.

Bar graphs are excellent for comparing the frequency of several categories at a glance. In our example, the 'K' responses would have the tallest bar, showing that "Do Not Know" was the most common answer.

Pie Chart

Pie charts are a pie-shaped circular graph used to illustrate the percentage breakdown of different responses. Each "slice" of the pie represents a category. The size of the slice is proportional to the percentage of responses for that category.
To create a pie chart based on our survey:

• Calculate the percentage of each response using relative frequency values.
• 'L' is 25%, 'N' is 30%, 'C' is 2.5%, and 'K' is 42.5%.
• Draw a circle and divide it into slices according to these percentages.

Pie charts provide a quick way to see the part-to-whole relationships, making it easy to understand the proportion of each response type. The bigger the slice, the higher the response rate for that category, making it visually straightforward which opinion was most or least chosen.