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Chapter 9: Problem 22

A USA Today/Gallup poll asked 1006 adult Americans how much it would bother them to stay in a room on the 13 th floor of a hotel. Interestingly, \(13 \%\) said it would bother them. The margin of error was 3 percentage points with \(95 \%\) confidence. Which of the following represents a reasonable interpretation of the survey results? For those not reasonable, explain the flaw. (a) We are \(95 \%\) confident that the proportion of adult Americans who would be bothered to stay in a room on the 13th floor is between 0.10 and 0.16 . (b) We are between \(92 \%\) and \(98 \%\) confident that \(13 \%\) of adult Americans would be bothered to stay in a room on the 13th floor. (c) In \(95 \%\) of samples of adult Americans, the proportion who would be bothered to stay in a room on the 13 th floor is between 0.10 and 0.16 . (d) We are \(95 \%\) confident that \(13 \%\) of adult Americans would be bothered to stay in a room on the 13 th floor.

Short Answer

Expert verified
Option (a) is the most reasonable interpretation.

Step by step solution

01

Understand Margin of Error and Confidence Interval

The margin of error is the range within which we expect the true population proportion to lie, with a given level of confidence. In this case, the margin of error is 3 percentage points and the confidence level is 95%. This means we can be 95% confident that the true proportion is within this margin of error around the sample proportion.
02

Calculate the Confidence Interval

Given that the sample proportion who would be bothered is 13% (or 0.13) and the margin of error is 3 percentage points (or 0.03), the confidence interval can be calculated as follows: Lower Bound = Sample Proportion - Margin of Error = 0.13 - 0.03 = 0.10 Upper Bound = Sample Proportion + Margin of Error = 0.13 + 0.03 = 0.16Therefore, the confidence interval is [0.10, 0.16].
03

Evaluate Each Option

(a) This option is reasonable because it directly states that we are 95% confident the true proportion is between 0.10 and 0.16.(b) This option is incorrect because it refers to the confidence level for a specific sample proportion rather than the interval within which we expect the true population proportion to lie.(c) This option is misleading because it suggests that 95% of all samples will have a proportion within this range, which is not the correct interpretation of a confidence interval.(d) This option is incorrect because it implies complete certainty about the sample proportion, which ignores the margin of error.
04

Select the Best Interpretation

Based on the above evaluations, option (a) is the most accurate interpretation of the survey results.

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Key Concepts

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

margin of error
In statistics, the margin of error represents the range within which we expect the true population parameter (like a proportion) to lie, given a particular level of confidence. It essentially provides a buffer around our sample estimate. For example, in the given poll, the margin of error is 3 percentage points. Hence, if 13% of the sample said it would bother them to stay on the 13th floor, the true proportion in the entire population likely falls between 10% and 16%. This range is our confidence interval.
confidence level
The confidence level in polling tells us how sure we can be that our margin of error contains the true population proportion. In this scenario, a 95% confidence level means we can be 95% certain that the real proportion of adult Americans who would be bothered by staying on the 13th floor falls within our calculated interval of 10% to 16%. It's important to remember that a higher confidence level leads to a wider margin of error, while a lower confidence level results in a narrower margin.
sample proportion
The sample proportion is the fraction of individuals in the surveyed sample who exhibited the trait or opinion being measured. In the poll described, 13% of the respondents said they would be bothered by staying in a room on the 13th floor. This 13% is our sample proportion. We use this value, along with the margin of error and confidence level, to infer the likely proportion in the entire population. But remember, the sample proportion is just an estimate, not a definitive measure of the population.
population proportion
The population proportion refers to the true proportion of the entire population that has the trait or opinion of interest. Unlike the sample proportion, it is usually unknown and estimated using the data we collect from our sample. In the example provided, while the sample proportion is 13%, the population proportion is presumed to be within the confidence interval of 10% to 16%, given the 95% confidence level and 3-point margin of error. We use the data from our sample to estimate this population parameter as accurately as possible.

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Most popular questions from this chapter

Construct the appropriate confidence interval. A simple random sample of size \(n=17\) is drawn from a population that is normally distributed. The sample mean is found to be \(\bar{x}=3.25,\) and the sample standard deviation is found to be \(s=1.17\). Construct a \(95 \%\) confidence interval for the population mean.

The American Society for Microbiology (ASM) and the Soap and Detergent Association (SDA) jointly commissioned two separate studies, both of which were conducted by Harris Interactive. In one of the studies, 1001 adults were interviewed by telephone and asked about their handwashing habits. In the telephone interviews, 921 of the adults said they always wash their hands in public restrooms. In the other study, the hand-washing behavior of 6076 adults was inconspicuously observed within public restrooms in four U.S. cities and 4679 of the 6076 adults were observed washing their hands. (a) In the telephone survey, what is the variable of interest? Is it qualitative or quantitative? (b) What is the sample in the telephone survey? What is the population to which this study applies? (c) Verify that the requirements for constructing a confidence interval for the population proportion of adults who say they always wash their hands in public restrooms are satisfied. (d) Using the results from the telephone interviews, construct a \(95 \%\) confidence interval for the proportion of adults who say they always wash their hands in public restrooms. (e) In the study where hand-washing behavior was observed, what is the variable of interest? Is it qualitative or quantitative? (f) We are told that 6076 adults were inconspicuously observed, but were not told how these adults were selected. We know randomness is a key ingredient in statistical studies that allows us to generalize results from a sample to a population. Suggest some ways randomness might have been used to select the individuals in this study. (g) Verify the requirements for constructing a confidence interval for the population proportion of adults who actually washed their hands while in a public restroom. (h) Using the results from the observational study, construct a \(95 \%\) confidence interval for the proportion of adults who wash their hands in public restrooms. (i) Based on your findings in parts (a) through (h), what might you conclude about the proportion of adults who say they always wash their hands versus the proportion of adults who actually wash their hands in public restrooms? (j) Cite some sources of variability in both studies.

Indicate whether a confidence interval for a proportion or mean should be constructed to estimate the variable of interest. Justify your response. Researchers at the Gallup Organization asked a random sample 1016 adult Americans aged 21 years or older, "Right now, do you think the state of moral values in the country as a whole is getting better, or getting worse?"

In a Gallup Poll, \(44 \%\) of the people polled answered "more strict" to the following question: "Do you feel that the laws covering the sale of firearms should be made more strict, less strict, or kept as they are now?" Suppose the margin of error in the poll was \(3.5 \%\) and the estimate was made with \(95 \%\) confidence. At least how many people were surveyed?

(a) Find the \(t\) -value such that the area in the right tail is 0.02 with 19 degrees of freedom. (b) Find the \(t\) -value such that the area in the right tail is 0.10 with 32 degrees of freedom. (c) Find the \(t\) -value such that the area left of the \(t\) -value is 0.05 with 6 degrees of freedom. [Hint: Use symmetry.] (d) Find the critical \(t\) -value that corresponds to \(95 \%\) confidence. Assume 16 degrees of freedom.
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