Question

Many people have misconceptions about how profitable small, consistent investments can be. In a survey of 1,010 randomly selected American 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 actual amount would be \(\$ 286,640\) ). Is there convincing evidence that less than \(40 \%\) of U.S. adults think that such an investment would result in a sum of over \(\$ 100,000\) ? Test the relevant hypotheses using \(\alpha=0.05\).

Short answer

There is insufficient evidence at the 0.05 level of significance to conclude that less than 40% of American adults believe that a regular $25 per week investment over 40 years with a 7% return would yield more than $100,000.

Step-by-step solution

Set up the hypothesis

The null hypothesis \(H_0\) is that 40% (or 0.4) of U.S adults think such an investment would yield over $100,000. Thus, \(H_0 : p = 0.4\). The alternative hypothesis \(H_a\) is that less than 40% of U.S. adults think so. Thus, \(H_a : p < 0.4\). p represents the population proportion.

Compute the sample proportion

The sample proportion \(\hat{p}\) is the number of successes (those who believe the investment yields over $100,000) divided by the total number in the sample. So, \(\hat{p} = \frac{374}{1010} = 0.37\).

Compute the Test Statistic

We compute the Z-score test statistic using the formula \(Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\), where n is the sample size. Substituting the appropriate values, \(Z = \frac{0.37 - 0.4}{\sqrt{\frac{0.4(1-0.4)}{1010}}}\), and calculating we find that the value of the Z statistic is approximately -1.36.

Find the P-value and make a decision

We now find the p-value corresponding to the observed value of the Z statistic. The p-value is the probability of observing a Z statistic as extreme as -1.36 or more, assuming the null hypothesis is true. From the standard normal distribution, the p-value is approximately 0.087. Since this p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis.

Draw the Conclusion

There is not enough evidence at the 0.05 level of significance to support the claim that less than 40% of U.S. adults believe that an investment of $25 per week over 40 years with a 7% annual return would result in a sum of over $100,000.

Key concepts

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement about the population that we assume to be true until proven otherwise. It's denoted as H₀ and in the context of our exercise, it asserts that at least 40% of U.S. adults believe that an investment of \(25 per week over 40 years with a 7% return would yield over \)100,000. Formally, we write it as H₀ : p = 0.4. Establishing a null hypothesis is critical as it provides a basis for comparison against our sample findings and helps in determining whether to support or reject it based on statistical analysis.

The null hypothesis is not a 'guess' but a presumption that there is no effect or no difference, and it's crucial to disprove it with evidence from our sample to assert that an alternative scenario is more plausible.

Alternative Hypothesis

Contrasting with the null hypothesis, we have the alternative hypothesis, represented as H_a or H₁. It serves as a statement that we believe to be true if the null hypothesis is rejected. For our survey problem, the alternative hypothesis proposes that less than 40% of U.S. adults think a \(25 weekly investment would grow beyond \)100,000 after 40 years. It is expressed mathematically as H_a : p < 0.4.

Alternative hypotheses pinpoint the expected direction of the expected outcome if the null hypothesis is deemed invalid. They are often what the research or question is aiming to demonstrate or support, making them pivotal to hypothesis testing.

Sample Proportion

The sample proportion, denoted as \(\bar{p}\bar{p}\bar{p}\) ), is a statistic that measures the fraction of individuals in a sample that have a particular attribute of interest. In the given problem, this proportion is the number of people who believe the investment exceeds $100,000 divided by the total sample size: \bar{p} = \frac{374}{1010} = 0.37. It is pivotal as a point estimate of the true population proportion. Understanding sample proportions allows us to use a group's response to infer insights about the overall population's beliefs or behaviors.

When we conduct tests such as a Z-test or T-test, the sample proportion helps build the test statistic which determines how far our sample outcome is from what is stipulated by the null hypothesis.

Test Statistic

The test statistic is a calculated value that falls within a known distribution and is used to assess the evidence against the null hypothesis. In the provided exercise, a Z-test is appropriate, and we calculate the Z-score as such: Z = \frac{\(\bar{p} - p\)}{\sqrt{\frac{p(1-p)}{n}}} where \(\bar{p}\) is the sample proportion, p is the null hypothesis proportion, and n is the sample size. This statistic tells us how many standard deviations our sample proportion is from the hypothesized population proportion.

By comparing our test statistic to a critical value or using it to determine a p-value, we can make informed decisions on whether to retain or reject our null hypothesis.

P-value

The p-value is a fundamental concept in hypothesis testing. It represents the probability of observing our sample results, or more extreme, considering the null hypothesis is true. In simpler terms, it answers the question, 'If the null hypothesis were true, how strange is our data?' For our example, the p-value is approximately 0.087. If this value is lower than our chosen significance level, we have enough evidence to reject the null hypothesis.

A common mistake is to misinterpret the p-value as the probability that the null hypothesis is true. Instead, it quantifies how extreme the observed data is under the null hypothesis - a small p-value indicates a higher incompatibility of the null hypothesis with the observed data.

Significance Level

The significance level, denoted by α, is a threshold chosen by the researcher to decide whether to reject the null hypothesis. It is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. In the example provided, a significance level of α = 0.05 is used. This means there is a 5% risk of concluding that a difference exists when there is none. It is a predefined parameter that researchers choose based on the desired confidence of their conclusions.

A smaller α value indicates a more stringent criterion for claiming a result to be statistically significant, thus reducing the risk of a Type I error. When the p-value exceeds the significance level, we fail to reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.