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Chapter 7: Problem 24

Increasing the sample size of an opinion poll will reduce the (a) bias of the estimates made from the data collected in the poll. (b) variability of the estimates made from the data collected in the poll. (c) effect of nonresponse on the poll. (d) variability of opinions in the sample. (e) variability of opinions in the population.

Short Answer

Expert verified
Increasing the sample size reduces the variability of the estimates, so the answer is (b).

Step by step solution

01

Understanding the Concept

In statistics, increasing the sample size of a poll generally impacts the precision of the estimates. The concept of variability is associated with the precision and reliability of statistical estimates. Let's break down how sample size affects these factors.
02

Defining Variability

Variability refers to how spread out the data points are in a data set or how much the estimates fluctuate when taking different samples. Larger samples help to stabilize and decrease this variability in estimates, making them more reliable.
03

Analyzing Provided Options

Review each option to determine which one is affected by increasing the sample size: - (a) Bias is systematic error that sample size alone cannot correct. - (b) Increasing the sample size reduces the variability (standard error) of the estimates. - (c) Nonresponse is a separate issue related to survey methods, not directly corrected by sample size. - (d) & (e) Variability of opinions is inherent to individual differences and the true population variability; larger samples do not change this intrinsic variance.
04

Choosing the Correct Answer

Based on the analysis, increasing the sample size will reduce the variability of the estimates made from the data collected in the poll. This is reflected in option (b).

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Key Concepts

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

Variability
In statistics, variability refers to how much the data points in a set differ from one another or from the overall mean. It is important because it gives us insight into the consistency or dispersion of the data.
  • High variability indicates that data points are spread out over a wide range of values.
  • Low variability means that the data points are closer to the mean.
Increasing the sample size in a study or poll generally reduces the variability of the statistical estimates. This is because larger samples tend to even out fluctuations and give a clearer picture of the population as a whole. When variability is minimized, the conclusions drawn from the data become more reliable. Variability can be quantitatively measured using statistical indicators like standard deviation or variance.
Bias in Statistics
Bias in statistics refers to a systematic error that leads to incorrect conclusions. Unlike variability, which is reduced with a larger sample size, bias is not affected by the number of samples.
  • It occurs when there is a consistent deviation from the true parameter.
  • Bias can stem from a poor study design or improper data collection methods.
For example, a biased sample might arise from a survey conducted only in urban areas but claiming to represent the entire population. Reducing bias requires careful planning and execution of the study to ensure that every segment of the population is properly represented. Understanding and minimizing bias is crucial for making accurate inference about the population.
Nonresponse in Polls
Nonresponse occurs when individuals selected for a survey or poll do not respond or participate. This can skew the results if the non-respondents differ significantly from those who do respond.
  • Nonresponse can create bias in the estimates because the sample may no longer be representative of the population.
  • Improving response rates through follow-ups or incentives can help mitigate this issue.
While increasing sample size may help with variability, it doesn't necessarily correct for nonresponse. Researchers need to employ strategies specifically targeted at encouraging higher participation. A poll with high nonresponse might miss key insights or portray results that don't truly reflect the population's opinions.
Statistical Estimates
Statistical estimates are values or metrics derived from sample data that aim to represent the broader population. Accurate estimates rely on both reduced variability and minimized bias.
  • Estimates often come in the form of means, proportions, or regression coefficients.
  • For more dependable estimates, increasing the sample size is beneficial as it tends to reduce the margin of error.
It鈥檚 essential that these estimates are calculated using proper statistical techniques to ensure their validity. By carefully designing the study and executing it with considerations for both variability and bias, we can produce trustworthy estimates. These concepts interlink to present a clearer and more accurate picture of the population being studied.

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Most popular questions from this chapter

Songs on an iPod David's iPod has about 10,000 songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of 225 seconds and a standard deviation of 60 seconds. Suppose we choose an SRS of 10 songs from this population and calculate the mean play time \(\bar{x}\) of these songs. What are the mean and the standard deviation of the sampling distribution of \(\bar{x}\) ? Explain.

Dead battery? A car company has found that the lifetime of its batteries varies from car to car according to a Normal distribution with mean \(\mu=48\) months and standard deviation \(\sigma=8.2\) months. The company installs a new brand of battery on an SRS of 8 cars. (a) If the new brand has the same lifetime distribution as the previous type of battery, describe the sampling distribution of the mean lifetime \(\bar{x}\). (b) The average life of the batteries on these 8 cars turns out to be \(\bar{x}=42.2\) months. Find the probability that the sample mean lifetime is 42.2 months or less if the lifetime distribution is unchanged. What conclusion would you draw?

Predict the election A polling organization plans to ask a random sample of likely voters who they plan to vote for in an upcoming election. The researchers will report the sample proportion \(\hat{p}\) that favors the incumbent as an estimate of the population proportion \(p\) that favors the incumbent. Explain to someone who knows little about statistics what it means to say that \(\hat{p}\) is an unbiased estimator of \(p\).

Who owns a Harley? Harley-Davidson motorcycles make up \(14 \%\) of all the motorcycles registered in the United States. You plan to interview an SRS of 500 motorcycle owners. How likely is your sample to contain \(20 \%\) or more who own Harleys? Show your work.

In a residential neighborhood, the median value of a house is 200,000 . For which of the following sample sizes is the sample median most likely to be above 250,000 ? \(\begin{array}{ll}\text { (a) } & n=10\end{array}\) (b) \(n=50\) \(\begin{array}{ll}\text { (c) } & n=100\end{array}\) (d) \(n=1000\) (e) Impossible to determine without more information.
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