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Chapter 9: Problem 50

A Gallup Poll found that \(59 \%\) of the people in its sample said "Yes" when asked, "Would you like to lose weight?" Gallup announced: "For results based on the total sample of national adults, one can say with \(95 \%\) confidence that the margin of (sampling) error is 卤3 percentage points." Does this interval provide convincing evidence that the actual proportion of U.S. adults who would say they want to lose weight differs from \(0.55 ?\) Justify your answer.

Short Answer

Expert verified
Yes, the interval does not include 0.55, suggesting the proportion differs.

Step by step solution

01

Understanding the Problem

We need to determine if the confidence interval for the sample proportion provides convincing evidence that the actual proportion of U.S. adults who want to lose weight differs from 0.55.
02

Determine the Confidence Interval

The sample proportion is given as 0.59 (or 59%) with a margin of error of 卤0.03 (or 卤3 percentage points). The confidence interval is calculated as follows: \[ (0.59 - 0.03, 0.59 + 0.03) = (0.56, 0.62) \] This means we are 95% confident that the true proportion of U.S. adults who want to lose weight lies between 0.56 and 0.62.
03

Compare with the Hypothesized Proportion

We compare the confidence interval \((0.56, 0.62)\) with the hypothesized proportion of 0.55. The interval does not include 0.55, which suggests evidence that the true proportion differs from 0.55.
04

Conclusion Based on Confidence Interval

Since the confidence interval does not include 0.55, we can conclude that there is convincing evidence the actual proportion of U.S. adults who would say they want to lose weight is different from 0.55.

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Key Concepts

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

Gallup Poll
The Gallup Poll is a well-known method for gauging public opinion by conducting random surveys of large populations. In this exercise, the Gallup Poll aimed to uncover what proportion of U.S. adults wished to lose weight. By asking a representative sample of people, the poll tries to reflect the views of all U.S. adults without surveying everyone individually.
Polling employs statistical sampling to save time and costs while still obtaining a reliable insight into the perspective of the broader public. In short, the Gallup Poll captures a snapshot of national sentiment on a given topic, such as weight loss desires in this case.
Sampling Error
Sampling error is a natural part of the survey process. It refers to the difference between the results from a sample and the results that would come from surveying the entire population.
In the context of the Gallup Poll, the sampling error was stated to be 卤3 percentage points.
This means that even though the poll found 59% of adults wanted to lose weight, the true proportion could reasonably be expected to fall anywhere between 56% and 62%, accounting for this error.
  • Sampling error is not a mistake, but rather a statistical probability that arises with any sample survey.
  • The margin of error helps establish a buffer zone, indicating the range in which the true population parameter lies, with a certain level of confidence.
  • For this poll, a 95% confidence level was chosen, meaning we can be 95% sure that the actual response rate falls within the stated range.
Sample Proportion
A sample proportion is the percentage of individuals in a sample with a certain characteristic, in this case, the desire to lose weight. The Gallup Poll found a sample proportion of 59%, meaning out of all the surveyed adults, 59% expressed a desire to shed some pounds. Sample proportion is crucial for making inferences about the entire population. Calculating a sample proportion helps estimate a range, known as the confidence interval, within which the true population proportion is likely to fall.
In studies such as these, understanding the sample proportion gives us an initial glimpse into broader population trends.
Hypothesized Proportion
The hypothesized proportion is a starting assumption or educated guess about a population parameter based on prior knowledge or claims. It's essentially the null hypothesis in statistical terms.
In the exercise, the hypothesized proportion was 0.55, or 55%, representing what one might expect the true proportion of U.S. adults wanting to lose weight to be.
  • The key question was whether the Gallup Poll's findings (and the derived confidence interval) support or refute this hypothesis.
  • In this instance, the confidence interval (0.56 to 0.62) did not include 0.55, indicating the observed data provides sufficient evidence to reject the hypothesized proportion.
  • This statistical reasoning is a vital part of drawing conclusions from survey data, as it helps determine if observed differences are meaningful or just due to random chance.

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Most popular questions from this chapter

(a) State hypotheses for a significance test to determine whether first responders are arriving within 8 minutes of the call more often. Be sure to define the parameter of interest. (b) Describe a Type I error and a Type II error in this setting and explain the consequences of each. (c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

You read that a statistical test at the \(\alpha=0.01\) level has probability 0.14 of making a Type II error when a specific alternative is true. What is the power of the test against this alternative?

A drug manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each batch of tablets produced is measured to control the compression process. The target value for the hardness is \(\mu=11.5 .\) The hardness data for a random sample of 20 tablets are \(\begin{array}{lllll}11.627 & 11.613 & 11.493 & 11.602 & 11.360 \\ 11.374 & 11.592 & 11.458 & 11.552 & 11.463 \\ 11.383 & 11.715 & 11.485 & 11.509 & 11.429 \\ 11.477 & 11.570 & 11.623 & 11.472 & 11.531\end{array}\) Is there convincing evidence at the \(5 \%\) level that the mean hardness of the tablets differs from the target value?

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