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Chapter 6: Problem 15

TV watching A social scientist uses the General Social Survey (GSS) to study how much time per day people spend watching TV. The variable denoted by TVHOURS at the GSS Web site measures this using the discrete values \(0,1,2, \ldots, 24\) a. Explain how, in theory, TV watching is a continuous random variable. b. An article about the study shows two histograms, both skewed to the right, to summarize TV watching for females and males. Since TV watching is in theory continuous, why were histograms used instead of curves? c. If the article instead showed two curves, explain what they would represent.

Short Answer

Expert verified
a) TV watching is continuous because it can take any fractional value. b) Histograms are used because real data is recorded in discrete units. c) Curves would represent probability density functions for continuous time distribution.

Step by step solution

01

Understanding TV Watching as Continuous

In theory, TV watching can be considered a continuous random variable because it is possible for individuals to watch TV for any fractional amount of time. Continuous random variables can take any value within a given range, and time can be measured as precisely as needed (e.g., 1.5 hours, 2.75 hours).
02

Justifying the Use of Histograms

Histograms are used instead of continuous curves because the actual data collected is discrete; that is, it is reported in whole hours and not fractions. Histograms provide a way to visualize the distribution of this categorical data, making it suitable for displaying a range of TV watching frequencies among different groups, such as males and females.
03

Interpreting Hypothetical Curves

If instead of histograms, the study had shown curves, these curves would represent probability density functions (PDFs) for the amounts of time spent watching TV by females and males. Such curves would model the continuous nature of TV watching and reflect the probability of individuals watching TV for any given amount of time within the possible range.

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Key Concepts

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

Continuous Random Variable
While TV watching was measured in whole hours in the study, it's theoretically a continuous random variable. This is because time can be divided into infinitely small units.
We can imagine that someone might watch TV for 1.5 hours or 2.7573 hours, just like how you can measure anything down to very precise levels.
  • Continuous random variables can take any value within a specified range, unlike discrete variables that have a limited number of distinct values.
  • Theoretical modeling of TV watching as continuous allows for more flexible and precise statistical analysis.
Discrete Data
In the context of the GSS dataset, TV watching was recorded as whole hours, thus making the data discrete. This means each recorded value is an integer, aligning with how data was practically collected rather than theorized.
  • Discrete data is characterized by numerical values that are separated by distinct gaps.
  • For instance, measuring something as whole hours narrows it down to discrete values like 0, 1, 2, etc.
This distinction helps when choosing the correct tools for data visualization and analysis.
Histograms
Histograms are perfect for representing discrete data like recorded TV hours.
Since the data has been collected in whole numbers, histograms capture this more appropriately than continuous curves.
  • Histograms display the frequency of each value or range of values in discrete data.
  • Each bar in a histogram represents the number of observations within the interval it covers.
  • With skewed histograms, like in the exercise's study, we can easily see how most responses might cluster at lower hours, tapering off toward higher hours.
Probability Density Function
While histograms showcase discrete data, if the study instead used curves, these would be a probability density function (PDF) showing TV watching as a continuous variable.
The curve visually represents the likelihood of different possibilities within the continuous time range of watching TV.
  • PDFs display the relative likelihood of a continuous random variable taking on a particular value.
  • The area under the curve within a specific range gives the probability of the random variable falling within that range.
  • In our exercise, if the study used PDFs for males and females separately, it would highlight the continuous nature of TV watching habits.
Understanding PDFs offers deeper insights into the data's distribution by recognizing the fluidity of human habits.

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Most popular questions from this chapter

Selling houses Let \(X\) represent the number of homes a real estate agent sells during a given month. Based on previous sales records, she estimates that \(\mathrm{P}(0)=0.68\), \(\mathrm{P}(1)=0.19, \mathrm{P}(2)=0.09, \mathrm{P}(3)=0.03, \mathrm{P}(4)=0.01\) with negligible probability for higher values of \(x\). a. Explain why it does not make sense to compute the mean of this probability distribution as \((0+1+2+3+4) / 5=2.0\) and claim that, on average, she expects to sell 2 homes. b. Find the correct mean and interpret.

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