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Chapter 7: Problem 58

A random sample of likely voters showed that \(49 \%\) planned to support Measure \(\mathrm{X}\). The margin of error is 3 percentage points with a \(95 \%\) confidence level. a. Using a carefully worded sentence, report the \(95 \%\) confidence interval for the percentage of voters who plan to support Measure \(X\). b. Is there evidence that Measure X will fail? c. Suppose the survey was taken on the streets of Miami and the measure was a Florida statewide measure. Explain how that would affect your conclusion.

Short Answer

Expert verified
The percentage of voters who plan to support Measure X is estimated to be between 46% and 52% with a 95% confidence level. Yes, there is a possibility that Measure X could fail. The conclusion might not be accurate if the survey was conducted only in Miami for a Florida statewide measure due to potential geographic and demographic biases.

Step by step solution

01

Describe the 95% Confidence Interval

The given data says that 49% of likely voters plan to support Measure X and the margin of error is 3%. This implies the confidence interval is \(49% \pm 3%\). So, the sentence describing the 95% confidence interval could be 'With a 95% confidence level, the percentage of voters who plan to support Measure X is likely between 46% and 52%'.
02

Assess the Likelihood of Measure X Failing

Based on the confidence interval (46% to 52%), the probability that Measure X will fail is likely if the lower limit of the interval is below 50%. In this case, the lower limit is 46% which is below 50%, implying that there is a possibility that Measure X could fail.
03

Discuss the Sampling Issues

If the survey was conducted only on the streets of Miami, but it concerns a Florida statewide measure, the conclusion about whether Measure X will pass or not might be skewed. This is because the data collected may not be representative of the entire population of Florida, but only specific to the population of Miami. Miami may have different demographics and political viewpoints than the rest of Florida, hence the conclusions derived might be not accurate.

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Key Concepts

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

Margin of Error
When we say there's a 'margin of error' in a survey or poll, it reflects the extent to which we can expect the survey results to differ from the true population value.

In the context of confidence intervals, imagine drawing a line in the sand 鈥� that line represents the percentage of people who support Measure X, according to our sample. The margin of error is like drawing a box around that line: we're saying we believe the true support is somewhere inside that box. Think of it as the wiggle room in our estimate. Here, a margin of error of 3 percentage points means if 49% of our sample supports Measure X, we're actually 95% confident that the support falls between 46% and 52% in the entire voting population.

Understanding the Range

It's crucial to realize that this range doesn't just capture likely outcomes; it also encapsulates our uncertainty. It's a reflection of random sampling error 鈥� the natural fluctuations that occur when we take a sample from a population. Keeping the margin of error modest is key to squaring our predictions with reality, ensuring we aren't caught off guard when election day rolls around.
Statistical Significance
Now, the concept of 'statistical significance' is a bit like asking the question: 'Could these results have happened by chance?' In our exercise, we can see that the 95% confidence interval falls both below and above the 50% mark in voter support for Measure X. So, what does this mean?

It indicates that while there's a chance Measure X could pass, there's also a statistically significant possibility it could fail, given that part of our interval is below the 50% threshold.

Decision Making with Confidence

For policymakers or campaign managers, this information is gold 鈥� it can help guide decisions about where to focus resources in the final run-up to the vote. If they can sway even a small percentage of voters, they might be able to push these numbers securely above the margin of error and ensure a win.
Representative Sample
The strength of any survey largely hinges on its 'representative sample.' Imagine trying to figure out the favorite ice cream flavor in a town by only asking people who come to one specific ice cream shop 鈥� you might end up with skewed results.

Similarly, in our exercise, if the survey for a Florida statewide measure was only conducted on the streets of Miami, it might not capture the full spectrum of Florida's diverse voter views. This could lead to an 'unrepresentative sample,' potentially leading us astray.

Broader Perspectives

To avoid this pitfall, it鈥檚 essential to draw samples from a variety of locations and demographics within Florida. A reliable survey must mirror the population it aims to represent, not just a slice of it. Otherwise, the confidence intervals we calculate might give false assurance, and we could find ourselves blindsided when the broader population weighs in on the measure.

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Most popular questions from this chapter

The website www.mlb.com compiles statistics on all professional baseball players. For the 2017 season, statistics were recorded for all 663 players. Of this population, the mean batting average was \(0.236\) with a standard deviation of \(0.064\). Would it be appropriate to use this data to construct a \(95 \%\) confidence interval for the mean batting average of professional baseball players for the 2017 season? If so, construct the interval. If not, explain why it would be inappropriate to do so.

The city of Chicago provides an open data set of the number of WiFi sessions at all of its public libraries. For 2014 , there were an average of \(451,846.9 \mathrm{WiFi}\) sessions per month at all Chicago public libraries. What is the correct notation for the value \(451,846.9 ?\)

Human blood is divided into 8 possible blood types. The rarest blood type is AB negative. Only \(1 \%\) of the population has this blood type. Suppose a random sample of 50 people is selected. Can we find the probability that more than \(3 \%\) of the sample has AB negative blood? If so, find the probability. If not, explain why this probability cannot be calculated.

Statistics student Hector Porath wanted to find out whether gender and the use of turn signals when driving were independent. He made notes when driving in his truck for several weeks. He noted the gender of each person that he observed and whether he or she used the turn signal when turning or changing lanes. (In his state, the law says that you must use your turn signal when changing lanes, as well as when turning.) The data he collected are shown in the table. $$ \begin{array}{|lcc|} \hline & \text { Men } & \text { Women } \\ \hline \text { Turn signal } & 585 & 452 \\ \hline \text { No signal } & 351 & 155 \\ \hline & 936 & 607 \\ \hline \end{array} $$ a. What percentage of men used turn signals, and what percentage of women used them? b. Assuming the conditions are met (although admittedly this was not a random selection), find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0, and explain whether this provides evidence that the proportions of men and women who use turn signals differ in the population. c. Another student collected similar data with a smaller sample size: $$ \begin{array}{|l|l|l|} \hline & \text { Men } & \text { Women } \\ \hline \text { Turn Signal } & 59 & 45 \\ \hline \text { No Signal } & 35 & 16 \\ \hline & 94 & 61 \\ \hline \end{array} $$ First find the percentage of men and the percentage of women who used turn signals, and then, assuming the conditions are met, find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0 , and explain whether this provides evidence that the percentage of men who use turn signals differs from the percentage of women who do so. d. Are the conclusions in parts \(\mathrm{b}\) and \(\mathrm{c}\) different? Explain.

The Centers for Disease Control and Prevention (CDC) conducts an annual Youth Risk Behavior Survey, surveying over 15,000 high school students. The 2015 survey reported that, while cigarette use among high school youth had declined to its lowest levels, \(24 \%\) of those surveyed reported using e-cigarettes. Identify the sample and population. Is the value \(24 \%\) a parameter or a statistic? What symbol would we use for this value?
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