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Chapter 3: Problem 9

In a 2010 survey, US teens aged \(12-18\) were asked what their favorite movie genre was. The results are shown below. \- Action: 351 \- Adventure: 171 \- Comedy: 651 \- Drama: 389 \- Horror: 287 \- Romance: 107 \- Undecided: 51 a. What is the implied population? b. How many people were sampled? c. What type of data is this? d. Create a relative frequency bar chart of the results. e. Create a pie chart of the results. f. Explain the advantages/disadvantages of the two charts. g. What is the statistic for the percentage of teens whose favorite movie genre is horror?

Short Answer

Expert verified
a. US teens aged 12-18. b. 2007 people. c. Categorical data. d/e. Draw the charts using relative frequencies. f. Bar charts compare categories; pie charts show parts of a whole. g. 14.3% prefer horror.

Step by step solution

01

Identify Implied Population

The implied population for this survey is the US teens aged 12-18 years old, which is the group from which the survey data was collected.
02

Calculate Sample Size

To find the sample size, sum all the survey responses. This includes: \[ 351 \text{ (Action)} + 171 \text{ (Adventure)} + 651 \text{ (Comedy)} + 389 \text{ (Drama)} + 287 \text{ (Horror)} + 107 \text{ (Romance)} + 51 \text{ (Undecided)} = 2007 \text{ teens} \].Therefore, 2007 people were sampled.
03

Determine Data Type

The data is categorical because it represents responses collected in categories (movie genres).
04

Calculate Relative Frequencies

To create a relative frequency bar chart, calculate the relative frequency for each genre by dividing the number for each genre by the total sample size:- Action: \( \frac{351}{2007} \approx 0.175 \)- Adventure: \( \frac{171}{2007} \approx 0.085 \)- Comedy: \( \frac{651}{2007} \approx 0.324 \)- Drama: \( \frac{389}{2007} \approx 0.194 \)- Horror: \( \frac{287}{2007} \approx 0.143 \)- Romance: \( \frac{107}{2007} \approx 0.053 \)- Undecided: \( \frac{51}{2007} \approx 0.025 \).
05

Create Bar Chart

Using the relative frequencies calculated in Step 4, draw a bar chart with each genre on the x-axis and the relative frequency on the y-axis. Each bar represents a movie genre.
06

Draw Pie Chart

For the pie chart, allocate each genre a sector proportionate to its relative frequency. This can be achieved by multiplying each relative frequency by 360掳 to find out the angle for each sector:- Action: \( 0.175 \times 360掳 \approx 63掳 \)- Adventure: \( 0.085 \times 360掳 \approx 31掳 \)- Comedy: \( 0.324 \times 360掳 \approx 117掳 \)- Drama: \( 0.194 \times 360掳 \approx 70掳 \)- Horror: \( 0.143 \times 360掳 \approx 51掳 \)- Romance: \( 0.053 \times 360掳 \approx 19掳 \)- Undecided: \( 0.025 \times 360掳 \approx 9掳 \).
07

Discuss Chart Advantages and Disadvantages

Bar charts are beneficial for easily comparing the size of each category clearly. However, they may not show the part-whole relationships as well as a pie chart. Pie charts are effective for showcasing the % contributions to a whole, yet they can be difficult to interpret accurately in terms of precise comparison.
08

Calculate Horror Genre Statistic

Calculate the percentage of teens who favor Horror by multiplying the relative frequency by 100. \[ 0.143 \times 100 = 14.3 ext{%} \].Thus, 14.3% of the teens surveyed indicated horror as their favorite movie genre.

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Key Concepts

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

Categorical Data
Categorical data represents information sorted into categories or groups. It's qualitative in nature, meaning it's more about describing qualities than providing numerical measurements. In the given exercise, the survey responses collected from US teens about their favorite movie genre are examples of categorical data. Each teen's choice falls into one of several distinguished categories like Action, Adventure, or Comedy.
  • Characteristics: There is no inherent order among categories. For example, one can't say that Comedy is "more" or "less" than Action unless a numerical value is assigned, like the number of votes.
  • Uses: Categorical data is typically represented visually using charts such as bar charts, pie charts, etc., that allow viewers to quickly compare the number of responses in each category.
  • Analysis: Simple descriptive statistics, like counts and percentages, are used to summarize categorical data. This helps us understand which categories are favored or less favored among the survey participants.
Sample Size Calculation
Sample size calculation in surveys is the process of determining how many individual responses or observations need to be collected to make meaningful and statistically valid conclusions. In the exercise, the sample size refers to the total number of survey respondents, which in this case was calculated by summing up the responses from all the categories.
  • Purpose: A proper sample size ensures that the data collected can reliably represent the larger population, minimizing errors and biases.
  • Calculation: From the exercise, sample size was calculated as follows:
    The total sample size is the sum of all responses:
    351 (Action) + 171 (Adventure) + 651 (Comedy) + 389 (Drama) + 287 (Horror) + 107 (Romance) + 51 (Undecided) = 2007.
    This means 2007 teens participated in the survey.
  • Considerations: When calculating sample sizes, researchers often consider factors like the desired confidence level and margin of error, which determine the precision and reliability of the survey results. Larger samples generally lead to more accurate results, but also require more resources.
Pie Chart
A pie chart is a circular statistical graphic used to represent data. Each "slice" of the pie illustrates a proportion of the whole. It's particularly useful for depicting percentage or proportional data, as each slice's angular size is proportionate to the quantity it represents.
  • How it works: To construct a pie chart, one calculates the angle for each category by multiplying its relative frequency (the proportion of the category relative to the entire dataset) by 360掳 as shown in the exercise.
    For example, for the Action genre:
    Relative frequency is 0.175, so the angle is approximately 63掳.
  • Advantages: Pie charts effectively show data as parts of a whole, offering a quick visual snapshot of the proportion each category contributes to the total. They are easily understood and can make the comparison between a small number of categories intuitive.
  • Disadvantages: In situations with many segments or where exact measurement is needed, pie charts can become confusing. Overly segmented data may be difficult to distinguish by eye, especially when the differences between categories are slight.

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Most popular questions from this chapter

A sports psychologist believes that people are more likely to be extroverted as an adult if they played team sports as a child. Describe two possible studies to test this theory. Design one as an observational study and the other as an experiment. Which is more practical?

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The city of Raleigh has 9,500 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 350 randomly selected registered voters was conducted. 112 said they'd vote for Brown, 207 said they'd vote for Feliz, and 31 were undecided. a. Who is the population of this survey? b. What is the size of the population? c. What is the size of the sample? d. Give the statistic for the percentage of voters surveyed who said they'd vote for Brown. e. If the margin of error was \(3.5 \%\), give the confidence interval for the percentage of voters surveyed that we might we expect to vote for Brown and explain what the confidence interval tells us.

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