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Chapter 8: Problem 104

A survey was conducted to determine what adults prefer in cell phone services. The results of the survey showed that \(73 \%\) of cell phone users wanted e-mail services. with a margin of error of \(\pm 4 \% .\) What is meant by the phrase "\pm4\%"? a. They estimate that \(4 \%\) of the surveyed population may change their minds between the time that the poll was conducted and the time that the results were published. b. There is a \(4 \%\) chance that the true percentage of cell phone users who want e-mail service will not be in the interval (0.69,0.77). c. Only \(4 \%\) of the population was surveyed. d. It would be unlikely to get the observed sample proportion of 0.73 unless the actual proportion of cell phone users who want e-mail service is between 0.69 and 0.77 e. The probability is. 04 that the sample proportion is in the interval (0.69,0.77)

Short Answer

Expert verified
The correct answer is d.

Step by step solution

01

Understanding the Concept

The term "margin of error" in a survey result is a statistical term that indicates the range within which the true population parameter (in this case, the percentage of adults who prefer email services) is expected to lie based on the sample data. The margin of error tells us how far off we might be from the actual population value.
02

Interpreting the Margin of Error

A margin of error of \(\pm 4\%\) means that the actual percentage of cell phone users who want e-mail services could be as much as 4\% higher or lower than the observed percentage from the survey. So, the true percentage is estimated to lie within the interval \(73\% - 4\% = 69\%\) and \(73\% + 4\% = 77\%\).
03

Evaluating the Options

Let's evaluate the provided options considering the interpretation of the margin of error: - Option (a) discusses changing minds, which is unrelated to margin of error. - Option (b) misinterprets the meaning of the margin of error as a probability of exclusion. - Option (c) incorrectly refers to the survey sample size. - Option (d) relates to the concept of interval estimation which correctly reflects that the sample proportion observed is consistent with the actual population proportion within the margin of error. - Option (e) incorrectly indicates a probability associated with the interval.
04

Choosing the Correct Option

From the analysis above, option (d) is correct. It reflects the statistical interpretation that the sample proportion of 0.73 would be unexpected unless the actual proportion lies between 0.69 and 0.77 (the range defined by the margin of error).

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Key Concepts

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

Survey Data Interpretation
Interpreting survey data is a critical skill in understanding statistics and making data-driven decisions. Surveys are a tool used to collect data from a specific sample that represents a larger population. In the context of our example, a survey was conducted to discover cell phone users' preferences regarding e-mail services. When interpreting survey data, one must consider the result alongside the margin of error.

The margin of error provides a range, ensuring there is always an acknowledgment of potential variation.
  • For instance, if 73% of respondents preferred e-mail services, the margin of error tells us how much trust we can place in this figure.
  • It indicates the expected range for the true proportion within which the population's actual views might lie.
Essentially, survey data interpretation takes into account both what the current sample states and how this might vary when applied to the larger population.
Confidence Interval
The concept of a confidence interval is fundamental in statistics and assures that the statistical analysis translates to real-world scenarios. When survey data indicates that 73% of respondents want e-mail service, and this is accompanied by a margin of error of \( \pm 4\% \), it constructs a confidence interval.
  • This interval goes from 69% to 77%, meaning that we are confident the true proportion of the entire population who prefer e-mail services falls within this range.
  • Confidence intervals give us an idea of the reliability of the survey results by illustrating the range where we believe the actual figure lies.
The width of the interval, dictated by the margin of error, can impact how we interpret data. Larger intervals may suggest less precision, while narrower intervals indicate a more precise survey result. Understanding these intervals helps us recognize the uncertainty inherent in sample-based measures.
Sample Proportion
The sample proportion is a cornerstone of statistical analysis, representing the ratio of individuals in a sample with a particular characteristic. In our case, the sample proportion is the 73% of surveyed adults who want e-mail services included in their phone plans.
  • This statistic is critical as it provides an estimate of the actual proportion in the larger population.
  • By combining the sample proportion with tools like the margin of error, we can gauge how representative this sample proportion is of the larger population.
It's important to remember that sample size and selection methods directly affect how much confidence we can place in the sample proportion. A well-executed survey aims to reflect the broader population, and the sample proportion is our primary tool in verifying this reflection.

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Most popular questions from this chapter

Exercise 8.129 suggests that \(S^{2}\) is superior to \(S^{\prime 2}\) in regard to bias and that \(S^{\prime 2}\) is superior to \(S^{2}\) because it possesses smaller variance. Which is the better estimator? [Hint: Compare the mean square errors.]

If \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from an exponential distribution with mean \(\theta\), then \(E\left(Y_{i}\right)=\theta\) and \(V\left(Y_{i}\right)=\theta^{2} .\) Thus, \(E(\bar{Y})=\theta\) and \(V(\bar{Y})=\theta^{2} / n,\) or \(\sigma_{\bar{Y}}=\theta / \sqrt{n} .\) Suggest an unbiased estimator for \(\theta\) and provide an estimate for the standard error of your estimator.

A factory operates with two machines of type \(A\) and one machine of type \(B\). The weekly repair costs \(X\) for type \(A\) machines are normally distributed with mean \(\mu_{1}\) and variance \(\sigma^{2}\). The weekly repair costs \(Y\) for machines of type \(B\) are also normally distributed but with mean \(\mu_{2}\) and variance \(3 \sigma^{2} .\) The expected repair cost per week for the factory is thus \(2 \mu_{1}+\mu_{2} .\) If you are given a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) on costs of type \(A\) machines and an independent random sample \(Y_{1}, Y_{2}, \ldots, Y_{m}\) on costs for type \(\mathrm{B}\) machines, show how you would construct a \(95 \%\) confidence interval for \(2 \mu_{1}+\mu_{2}\) a. if \(\sigma^{2}\) is known. b. if \(\sigma^{2}\) is not known.

A random sample of size 25 was taken from a normal population with \(\sigma^{2}=6\). A confidence interval for the mean was given as \((5.37,7.37) .\) What is the confidence coefficient associated with this interval?

The following statistics are the result of an experiment conducted by P. I. Ward to investigate a theory concerning the molting behavior of the male Gammarus pulex, a small crustacean. \(^{\star}\) If a male needs to molt while paired with a female, he must release her, and so loses her. The theory is that the male \(G\). pulex is able to postpone molting, thereby reducing the possibility of losing his mate. Ward randomly assigned 100 pairs of males and females to two groups of 50 each. Pairs in the first group were maintained together (normal); those in the second group were separated (split). The length of time to molt was recorded for both males and females, and the means, standard deviations, and sample sizes are shown in the accompanying table. (The number of crustaceans in each of the four samples is less than 50 because some in each group did not survive until molting time.) a. Find a \(99 \%\) confidence interval for the difference in mean molt time for "normal" males versus those "split" from their mates. b. Interpret the interval.
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