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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

According to a survey of 1000 adult Americans conducted by Opinion Research Corporation, 210 of those surveyed said playing the lottery would be the most practical way for them to accumulate \(\$ 200,000\) in net wealth in their lifetime ("One in Five Believe Path to Riches Is the Lottery," San Luis Obispo Tribune, January 11,2006 ). Although the article does not describe how the sample was selected, for purposes of this exercise, assume that the sample can be regarded as a random sample of adult Americans. Is there convincing evidence that more than \(20 \%\) of adult Americans believe that playing the lottery is the best strategy for accumulating \(\$ 200,000\) in net wealth?

Much concern has been expressed in recent years regarding the practice of using nitrates as meat preservatives. In one study involving possible effects of these chemicals, bacteria cultures were grown in a medium containing nitrates. The rate of uptake of radio-labeled amino acid was then determined for each culture, yielding the following observations: \(\begin{array}{llllllll}7251 & 6871 & 9632 & 6866 & 9094 & 5849 & 8957 & 7978 \\\ 7064 & 7494 & 7883 & 8178 & 7523 & 8724 & 7468 & \end{array}\) Suppose that it is known that the true average uptake for cultures without nitrates is 8000 . Do the data suggest that the addition of nitrates results in a decrease in the true average uptake? Test the appropriate hypotheses using a significance level of \(.10\).

The true average diameter of ball bearings of a certain type is supposed to be \(0.5\) in. What conclusion is appropriate when testing \(H_{0}: \mu=0.5\) versus \(H_{a}: \mu \neq 0.5 \mathrm{in}\) each of the following situations: a. \(n=13, t=1.6, \alpha=.05\) b. \(n=13, t=-1.6, \alpha=.05\) c. \(n=25, t=-2.6, \alpha=.01\) d. \(n=25, t=-3.6\)

A county commissioner must vote on a resolution that would commit substantial resources to the construction of a sewer in an outlying residential area. Her fiscal decisions have been criticized in the past, so she decides to take a survey of constituents to find out whether they favor spending money for a sewer system. She will vote to appropriate funds only if she can be fairly certain that a majority of the people in her district favor the measure. What hypotheses should she test?

Pizza Hut, after test-marketing a new product called the Bigfoot Pizza, concluded that introduction of the Bigfoot nationwide would increase its sales by more than \(14 \%\) (USA Today, April 2, 1993). This conclusion was based on recording sales information for a random sample of Pizza Hut restaurants selected for the marketing trial. With \(\mu\) denoting the mean percentage increase in sales for all Pizza Hut restaurants, consider using the sample data to decide between \(H_{0}: \mu=14\) and \(H_{a}: \mu>14\). a. Is Pizza Hut's conclusion consistent with a decision to reject \(H_{0}\) or to fail to reject \(H_{0}\) ? b. If Pizza Hut is incorrect in its conclusion, is the company making a Type I or a Type II error?
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