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Chapter 7: Problem 18

Predict the election Just before a presidential election, a national opinion poll increases the size of its weekly random sample from the usual 1500 people to 4000 people. (a) Does the larger random sample reduce the bias of the poll result? Explain. (b) Does it reduce the variability of the result? Explain.

Short Answer

Expert verified
(a) No, larger samples do not reduce bias. (b) Yes, larger samples reduce variability.

Step by step solution

01

Understanding Bias

Bias refers to systematic errors that can affect the accuracy of poll results. It is primarily influenced by how the sample is selected or how questions are phrased, not by the sample size. Hence, increasing the sample size from 1500 to 4000 does not inherently reduce bias because bias is related to the methodology of sampling rather than the number of participants.
02

Understanding Variability

Variability refers to the extent to which poll results would differ if the poll was repeated multiple times. It is affected by the size of the sample 鈥� larger samples tend to have lower variability. Increasing the sample size from 1500 to 4000 reduces the variability of the poll results, as larger samples provide a more accurate estimate of the population parameter by reducing the margin of error.
03

Conclusion on Bias and Variability

In summary, increasing the sample size does not reduce the bias of the poll result, as bias is related to the sampling method. However, it does reduce variability as a larger sample size decreases the margin of error and provides more reliable results.

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Key Concepts

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

Bias in Statistics
When discussing statistics, bias is a critical concept that impacts the trustworthiness of the data collected. Bias occurs when there is a systematic error in the sampling process or in the data collection methodology. This systematic error means that the sample results do not accurately reflect the true situation, which can mislead conclusions or predictions. For example, if a poll consistently over-represents one demographic group, the results will be biased in favor of that group. This is not corrected by merely increasing sample size. Instead, eliminating bias requires improving the sampling method or ensuring the data collection process is neutral. To minimize bias, focus on ensuring that sampling methods are truly random and representative of the population.
Sample Size and Variability
In statistics, sample size and variability are interconnected. Variability refers to how much the results would differ if the study were repeated multiple times. It reflects the potential variations in outcome due to sample differences. A smaller sample size can lead to greater variability, meaning results can swing dramatically each time the study is conducted. Increasing the sample size, like in the election poll from 1500 to 4000 participants, tends to reduce this variability. Larger samples provide a better approximation of the population鈥檚 true parameter. With less variability, the results are more consistent and reliable.
Margin of Error
The margin of error is a statistical measure that gives an estimate of the range within which the actual population parameter lies. It is closely linked with both sample size and variability. A larger sample size reduces the margin of error, thereby increasing the precision of the survey results. For instance, by expanding a sample size from 1500 to 4000, the margin of error decreases, leading to tighter confidence intervals and a higher level of confidence in the findings. This makes the polling data not only more precise but also more actionable when predicting outcomes such as election results. Always aim for an optimal sample size to balance cost and precision effectively.

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Most popular questions from this chapter

If we take a simple random sample of size \(n=500\) from a population of size \(5,000,000,\) the variability of our estimate will be (a) much less than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (b) slightly less than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (c) about the same as the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (d) slightly greater than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) . (e) much greater than the variability for a sample of size \(n=500\) from a population of size \(50,000,000\) .

Bad carpet The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.2 flaws per square yard. The population distribution cannot be Normal, because a count takes only whole-number values. An inspector studies 200 square yards of the material, records the number of flaws found in each square yard, and calculates \(\overline{x}\) , the mean number of flaws per square yard inspected. Find the probability that the mean number of flaws exceeds 2 per square yard. Follow the four-step process.

Do you go to church? The Gallup Poll asked a random sample of 1785 adults whether they attended church or synagogue during the past week. Of the respondents, 44\(\%\) said they did attend. Suppose that 40\(\%\) of the adult population actually went to church or synagogue last week. Let \(\hat{p}\) be the proportion of people in the sample who attended church or synagogue. (a) What is the mean of the sampling distribution of \(\hat{p}\) ? Why? (b) Find the standard deviation of the sampling distribution of \(\hat{p} .\) Check to see if the 10\(\%\) condition is met. (c) Is the sampling distribution of \(\hat{p}\) approximately Normal? Check to see if the Normal condition is met. (d) Find the probability of obtaining a sample of 1785 adults in which 44\(\%\) or more say they attended church or synagogue last week. Do you have any doubts about the result of this poll?

For each boldface number in Exercises 5 to \(8,(1)\) state whether it is a parameter or a statistic and (2) use appropriate notation to describe each number; for example, \(p=0.65\) Unlisted numbers A telemarketing firm in Los Angeles uses a device that dials residential telephone numbers in that city at random. Of the first 100 numbers dialed, 48\(\%\) are unlisted. This is not surprising because 52\(\%\) of all Los Angeles residential phones are unlisted.

Scooping beads A statistics teacher fills a large container with 1000 white and 3000 red beads and then mixes the beads thoroughly. She then has her students take repeated SRSs of 50 beads from the container. After many SRSs, the values of the sample proportion \(\hat{p}\) of red beads are approximated well by a Normal distribution with mean 0.75 and standard deviation 0.06 . (a) What is the population? Describe the population distribution. (b) Describe the sampling distribution of \(\hat{p} .\) How is it different from the population distribution?
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