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Chapter 10: Problem 31

In a survey conducted by Yahoo Small Business, 1432 of 1813 adults surveyed said that they would alter their shopping habits if gas prices remain high (Associated Press, November 30,2005 ). The article did not say how the sample was selected, but for purposes of this exercise, assume that it is reasonable to regard this sample as representative of adult Americans. Based on these survey data, is it reasonable to conclude that more than three-quarters of adult Americans plan to alter their shopping habits if gas prices remain high?

Short Answer

Expert verified
Yes, based on this survey data, it is reasonable to conclude that more than three-quarters of adult Americans would alter their shopping habits if gas prices remain high.

Step by step solution

01

Compute the Sample Proportion

We first need to compute the sample proportion (p虃), which is an estimate of the population proportion (p). We do this by dividing the number of affirmative responses (1432) by the total number of responses (1813).
02

Compute the Sample Proportion

After doing the calculation from step 1, we get \(p虃 = 0.79\) (rounded to two decimal places). This shows that around 79% of the sample (and hence, theoretically the population) would alter their shopping habits if gas prices remain high.
03

Interpret the Results

The sample proportion of 0.79 (or 79%) is greater than 0.75 (or three-quarters). This means, based on the survey data, it is quite plausible to infer that more than three-quarters of all adult Americans plan to change their shopping habits if gas prices stay high.

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Key Concepts

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

Sample Proportion
The sample proportion is a way of estimating what the whole population might do, based on a smaller group. In the survey you are looking at, 1432 out of 1813 adults reported they would change their shopping habits due to high gas prices. To find the sample proportion, simply divide the number of people who said they would change their habits by the total number surveyed. So, the calculation is: \[ \hat{p} = \frac{1432}{1813} \approx 0.79\]This tells you that about 79% of the people surveyed said they would change their habits.This sample proportion is a quick estimate to tell you who, out of this small group, might think the same way in the entire group of all adults.
Population Proportion
The population proportion represents the true value of a characteristic in the entire group, such as all adults in this case. But it's mostly unknown unless you talk to everyone! However, you can estimate it using the sample proportion. If you recall, the sample proportion found was 0.79 or 79%. It suggests that in the entire U.S. adult population, a similar percentage might alter their shopping behaviors. However, remember that your sample proportion is just an estimate. It can give you a good idea, but it might not perfectly reflect the population because it depends on how the sample was chosen.
Confidence Level
The confidence level tells you how sure you can be about your estimates. It's usually expressed as a percentage, like 90% or 95%. This percentage shows how often you would expect the result to fall within a certain range if you repeated the survey many times. With a 95% confidence level, you think, "There's a 95% chance our sample estimate is catching the true population proportion." A higher confidence level means you need a wider range to be sure the estimate is accurate. If the estimation from the survey shows that more than 75% of the population might act a certain way, using a confidence level helps determine how sure you are about that prediction. Confidence levels play a crucial role in interpreting survey results and making informed decisions.

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Most popular questions from this chapter

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