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Chapter 16: Problem 27

Confidence intervals Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true? a. For a given sample size, higher confidence means a smaller margin of error. b. For a specified confidence level, larger samples provide smaller margins of error. c. For a fixed margin of error, larger samples provide greater confidence. d. For a given confidence level, halving the margin of error requires a sample twice as large.

Short Answer

Expert verified
Statement a is false. Statement b and statement c are true. Statement d is false.

Step by step solution

01

Assess Statement a

This statement says: 'for a given sample size, higher confidence means a smaller margin of error.' This is incorrect. Increasing the confidence level, while keeping the sample size constant, actually leads to a larger margin of error. This is because with higher confidence, the interval needs to widen to ensure that the true population value is included.
02

Assess Statement b

This statement says: 'for a specified confidence level, larger samples provide smaller margins of error.' This is correct. If the confidence level is held constant, increasing the sample size decreases the margin of error. This is because a larger sample provides more 'information' about the population, and thus the estimate is made with greater precision.
03

Assess Statement c

This statement says: 'for a fixed margin of error, larger samples provide greater confidence.' This is correct. With a larger sample size, the estimate of the population parameter is more precise, so for a set margin of error, a larger sample size would allow for a higher level of confidence that the true population parameter is within the specified interval.
04

Assess Statement d

This statement says: 'for a given confidence level, halving the margin of error requires a sample twice as large.' This is incorrect. In fact, because the margin of error is inversely proportional to the square root of the sample size, a four-fold increase in sample size (not a two-fold) would be required to halve the margin of error for a given confidence level.

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Key Concepts

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

Margin of Error
The margin of error (MOE) plays a crucial role in understanding how closely the results of your sample might match the entire population. It helps us gauge the range within which we can expect the true population parameter to fall. The smaller the margin of error, the closer we can expect our sample statistic to be to the real population value.

It's important to note that the margin of error is affected by both the sample size and the level of confidence. A common misconception, highlighted in the exercise, is that a higher level of confidence will reduce the margin of error, when in fact it does the opposite. A higher level of confidence will result in a larger margin of error, all else being equal, because we're widening our net to be more certain that the population parameter is captured within our interval.
Sample Size
Sample size significantly affects the precision of our estimates and thus the margin of error. To put it simply: the larger the sample size, the smaller the margin of error for a given level of confidence. This relationship is really well exemplified in the second statement of the exercise.

As a rule of thumb, increasing the sample size improves the accuracy of our estimates because it's more representative of the population. However, it's vital to realize that the sample size doesn't increase proportionally to the desired decrease in the margin of error. To halve the margin of error, one would need to quadruple the sample size, not just double it. This underscores the principle that precision increases with the square root of the sample size.
Level of Confidence
The level of confidence is expressed as a percentage and represents how certain we are that the true population parameter lies within our calculated confidence interval. Common confidence levels include 90%, 95%, and 99%.

A critical aspect of confidence intervals not to be confused is that while increasing the level of confidence results in a broader interval, it doesn't necessarily mean better data or more accuracy. Increasing the level of confidence means accepting a larger margin of error in exchange for increased assurance that the interval contains the true population value. It's also worth noting that the third statement from our exercise effectively captures the relationship between sample size and the level of confidence - a larger sample size can give us greater confidence without altering the margin of error, making our estimate more trustworthy.

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Most popular questions from this chapter

Safe food Some food retailers propose subjecting food to a low level of radiation in order to improve safety, but sale of such "irradiated" food is opposed by many people. Suppose a grocer wants to find out what his customers think. He has cashiers distribute surveys at checkout and ask customers to fill them out and drop them in a box near the front door. He gets responses from 122 customers, of whom 78 oppose the radiation treatments. What can the grocer conclude about the opinions of all his customers?

How's life? Gallup regularly conducts a poll using a "Cantril scale," which asks respondents to imagine a ladder with 10 rungs. Rung 0 represents the worst possible life, and rung 10 represents the best possible life. Respondents are asked what rung they would say they are on. Responses are classified as "Thriving" (standing on rung 7 or higher, and expecting to be on rung 8 or higher five years from now), "Suffering" (standing on rung 4 or lower and expecting to be on rung 4 or lower five years from now), or "Struggling" (not thriving or suffering). In the first half of \(2016,\) Gallup found that the index had reached a new high of \(55.7 \%\) thriving with a sample size of 105,000 . (www.gallup.com/poll/194816/ americans-life- evaluations-improve-during-obama-era.aspx) a. Explain the meaning of \(\hat{p}=0.557\) in the context of this situation. b. Calculate the standard error of \(\hat{p}\). c. Explain what this standard error means in the context of this situation.

Baseball fans In a poll taken in December 2012, Gallup asked 1006 national adults whether they were baseball fans; \(48 \%\) said they were. Almost five years earlier, in February \(2008,\) only \(35 \%\) of a similar-size sample had reported being baseball fans. a. Find the margin of error for the 2012 poll if we want \(90 \%\) confidence in our estimate of the percent of national adults who are baseball fans. b. Explain what that margin of error means. c. If we wanted to be \(99 \%\) confident, would the margin of error be larger or smaller? Explain. d. Find that margin of error. e. In general, if all other aspects of the situation remain the same, will smaller margins of error produce greater or less confidence in the interval?

31\. Mislabeled seafood In 2013 the environmental group Oceana (usa.oceana.org) analyzed 1215 samples of seafood purchased across the United States and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that \(33 \%\) of the seafood was mislabeled according to U.S. Food and Drug Administration guidelines. a. Construct a \(95 \%\) confidence interval for the proportion of all seafood sold in the United States that is mislabeled or misidentified. b. Explain what your confidence interval says about seafood sold in the United States. c. A 2009 report by the Government Accountability Office says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 1215 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

Rickets Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated in England during the 1950 s, but now there is concern that a generation of children more likely to watch TV or play computer games than spend time outdoors is at increased risk. A recent study of 2700 children randomly selected from all parts of England found \(20 \%\) of them deficient in vitamin D. a. Find a \(98 \%\) confidence interval. b. Explain carefully what your interval means. c. Explain what "98\% confidence" means.
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