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Chapter 10: Problem 77

Many people have misconceptions about how profitable small, consistent investments can be. In a survey of 1010 randomly selected U.S. adults (Associated Press, October 29,1999 ), only 374 responded that they thought that an investment of \(\$ 25\) per week over 40 years with a \(7 \%\) annual return would result in a sum of over \(\$ 100,000\) (the correct amount is \(\$ 286,640\) ). Is there sufficient evidence to conclude that less than \(40 \%\) of U.S. adults are aware that such an investment would result in a sum of over \(\$ 100,000 ?\) Test the relevant hypotheses using \(\alpha=.05\).

Short Answer

Expert verified
Depending on whether the calculated \(Z\) is less than the \(Z_{critical}\), we either reject or do not reject the null hypothesis.

Step by step solution

01

State the Hypotheses

The null hypothesis is that \(40 \%\) or more of U.S adults are aware of the investment result, that is \(p \geq 0.40\), where \(p\) is the population proportion. The alternative hypothesis is what we're testing for, that less than \(40 \%\) are aware, that is \(p < 0.40\).
02

Choose an appropriate Test Statistic

For a hypothesis test involving a population proportion, we use a z-test. The formula for the test statistic is \[Z = \frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\] where \(\hat{p}\) is the sample proportion, \(p_0\) is the population proportion from the null hypothesis and \(n\) is the sample size.
03

Calculate

From the question, we have \(\hat{p} = \frac{374}{1010}\) and \(n = 1010\). Substituting in the above formula, we get the test statistic \(Z\).
04

Determining the Critical Value

Since it's a left-tailed test (evidence for less than) and \(\alpha = 0.05\), we find the critical value \(Z_{critical}\) from the standard normal table corresponding to area \(0.05\). If our test statistic \(Z\) is less than to \(Z_{critical}\), then we reject the null hypothesis
05

Reaching the conclusion

Based on the above computations, if our \(Z\) (test statistic) is less than \(Z_{critical}\)(critical value), then we reject the null hypothesis in favour of the alternative hypothesis. Basically, if we reject the null hypothesis, then there is enough evidence to believe that less than \(40\%\) of U.S adults are aware that such an investment would result in a sum of over \$100,000. Otherwise, there is not enough evidence.

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Key Concepts

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

Population Proportion
When we talk about population proportion in the context of hypothesis testing, we are referring to the fraction or percentage of the entire group that exhibits a particular characteristic. For example, in our survey of U.S. adults, the population proportion represents the percentage who are aware that a small investment can grow significantly over time. Understanding this helps us estimate how a small sample's result can reflect the entire population's behavior. In this example, we want to see if less than 40% of people are aware of the investment result.
Z-Test
A Z-Test is a statistical method used to determine if there is a significant difference between sample data and a known population. It's particularly useful in hypothesis testing for means and proportions when the population variance is known or the sample is large. In our case, the Z-Test formula \[Z = \frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\] is used to compare the sample proportion \(\hat{p}\) with the hypothesized population proportion \(p_0\). This test helps us decide whether to accept or reject the null hypothesis. By calculating the Z value, we can determine if our sample proportion significantly deviates from what the null hypothesis claims.
Null Hypothesis
The null hypothesis (denoted as \(H_0\)) is a statement we assume to be true at the start of a hypothesis test. It represents no change or no difference. In this scenario, our null hypothesis is that 40% or more of the population is aware of the investment growth.With \(H_0: p \geq 0.40\), we suggest that things are either as expected or better in terms of awareness. If the data provides enough evidence against this hypothesis, we may reject it and consider the alternative hypothesis instead.
Alternative Hypothesis
The alternative hypothesis (denoted as \(H_1\)) is what we consider if the null hypothesis is rejected. It usually represents a new effect or difference that we are trying to detect.In our investment awareness example, the alternative hypothesis is \(H_1: p < 0.40\). This means we are testing to find out if less than 40% of the population is aware of the returns from the investment. The alternative hypothesis points to the specific direction of the effect we are looking to demonstrate with our data.

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Most popular questions from this chapter

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?

The article "Poll Finds Most Oppose Return to Draft, Wouldn't Encourage Children to Enlist" (Associated Press, December 18,2005 ) reports that in a random sample of 1000 American adults, 700 indicated that they oppose the reinstatement of a military draft. Is there convincing evidence that the proportion of American adults who oppose reinstatement of the draft is greater than twothirds? Use a significance level of \(.05\).

Typically, only very brave students are willing to speak out in a college classroom. Student participation may be especially difficult if the individual is from a different culture or country. The article "An Assessment of Class Participation by International Graduate Students" (Journal of College Student Development [1995]: 132- 140) considered a numerical "speaking-up" scale, with possible values from 3 to 15 (a low value means that a student rarely speaks). For a random sample of 64 males from Asian countries where English is not the official language, the sample mean and sample standard deviation were \(8.75\) and \(2.57\), respectively. Suppose that the mean for the population of all males having English as their native language is \(10.0\) (suggested by data in the article). Does it appear that the population mean for males from non-English-speaking Asian countries is smaller than \(10.0 ?\)

A student organization uses the proceeds from a particular soft-drink dispensing machine to finance its activities. The price per can had been \(\$ 0.75\) for a long time, and the average daily revenue during that period had been \(\$ 75.00\). The price was recently increased to \(\$ 1.00\) per can. A random sample of \(n=20\) days after the price increase yielded a sample average daily revenue and sample standard deviation of \(\$ 70.00\) and \(\$ 4.20\), respectively. Does this information suggest that the true average daily revenue has decreased from its value before the price increase? Test the appropriate hypotheses using \(\alpha=.05\).

White remains the most popular car color in the United States, but its popularity appears to be slipping. According to an annual survey by DuPont (Los Angeles Times, February 22,1994 ), white was the color of \(20 \%\) of the vehicles purchased during 1993 , a decline of \(4 \%\) from the previous year. (According to a DuPont spokesperson, white represents "innocence, purity, honesty, and cleanliness.") A random sample of 400 cars purchased during this period in a certain metropolitan area resulted in 100 cars that were white. Does the proportion of all cars purchased in this area that are white appear to differ from the national percentage? Test the relevant hypotheses using \(\alpha=.05\). Does your conclusion change if \(\alpha=.01\) is used?
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