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Chapter 8: Problem 44

Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal. Define the random variable \(X\) in words.

Short Answer

Expert verified
The random variable \(X\) represents the number of hours an American watches television in a month.

Step by step solution

01

Identify the Context

We have been given a scenario where a group of Americans has been surveyed to find out their average television watching time. This context involves a statistical study where variables represent certain quantities being measured.
02

Understand the Random Variable

In statistical terms, a random variable is a numeric outcome that arises due to a random process, in this case, the process of sampling Americans and measuring their television watching habits each month.
03

Define the Random Variable

Since we're measuring the 'number of hours watched' by each individual surveyed on a monthly basis, the random variable will represent this particular measurement across different individuals.

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Key Concepts

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

Survey Data Analysis
Survey data analysis is the task of interpreting data collected from a survey, transforming raw numbers into meaningful information. This exercise involves analyzing data from a survey where 108 Americans reported their TV watching habits. To make sense of the data, it's essential to summarize what the average respondent does. The respondents watched an average of 151 hours of TV each month. However, while averages are useful, they don't tell the whole story.
  • The objective of survey data analysis is to draw conclusions that apply to a broader population, not just the sample.
  • Collecting data from a carefully chosen group helps ensure that findings can be generalized to the entire population.
  • A key aim is to identify patterns, trends, and insights from the survey results to make data-driven decisions.
By understanding how to analyze survey data, you can accurately interpret what it means when the random variable, such as hours watched, is described by specific statistics like the mean.
Standard Deviation
Standard deviation is a crucial statistic that measures how spread out the numbers are in a data set. In this context, it reflects how much individual respondents' TV-watching hours deviate from the average of 151 hours. For our example, the standard deviation came out to be 32 hours. This number has several important implications:
  • If the standard deviation is small, most data points are close to the average. A large standard deviation indicates that data points are more spread out.
  • For example, a deviation of 32 hours suggests there's significant variability in how much time people spend watching TV.
  • Standard deviation helps in identifying the range and consistency of behaviors or characteristics within a data set.
Understanding the standard deviation allows us to get a clearer image of data reliability and the dispersion of hours watched among individuals.
Normal Distribution
The normal distribution, often referred to as a bell curve, is a fundamental concept in statistics. It's called so because of its bell shape and its ubiquity in representing real-world data. The survey in question assumes that TV-watching habits of Americans are normally distributed. This assumption means:
  • Most data points (here, hours of TV watched) cluster around the mean (151 hours).
  • The probability of extreme deviations from the mean is low, following a symmetrical distribution on both sides.
  • Approximately 68% of data values lie within one standard deviation, and about 95% lie within two standard deviations from the mean, under a normal distribution.
In any analysis, assuming a normal distribution helps in making predictions and conducting statistical tests. It simplifies the complex nature of data to fit into a model that is widely understood and used in various statistical analyses.

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Most popular questions from this chapter

Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal. If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?

Use the following information to answer the next five exercises. Suppose the marketing company did do a survey. They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. We are interested in the population of households where women make the majority of the purchasing decisions. Define the random variables \(X\) and \(P^{\prime}\) in words.

Construct a 95\(\%\) Confidence Interval for the true mean age of winter Foothill College students by working out then answering the next seven exercises. Using the same mean, standard deviation, and level of confidence, suppose that n were 69 instead of 25. Would the error bound become larger or smaller? How do you know?

Use the following information to answer the next five exercises. Suppose the marketing company did do a survey. They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. We are interested in the population of households where women make the majority of the purchasing decisions. Identify the following: a. \(x=\) _____ b. \(n=\) ______ c. \(p^{\prime}=\) _____

Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal. Suppose the Census needed to be 98% confident of the population mean length of time. Would the Census have to survey more people? Why or why not?
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