Question

According to a survey of 1,000 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 is a random sample of adult Americans. Suppose that you want to use the data from this survey to decide if there is 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. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is 0.215 . What conclusion would you reach if \(\alpha=0.05 ?\)

Short answer

a. The hypotheses are \(H_0: p = 0.20\) and \(H_A: p > 0.20\).\nb. Given that the p-value is greater than the level of significance (\(\alpha\)), we fail to reject the null hypothesis. Therefore, there's not enough evidence to conclude that more than 20% of adult Americans believe that playing the lottery is the best strategy to accumulate $200,000 in net wealth.

Step-by-step solution

Formulate Hypotheses

Hypotheses are statements about population parameters that we want to test. In this case, the null hypothesis \((H_0)\) assumes no change from the status quo (20%). The alternative hypothesis \((H_A)\) reflects the change we're interested in – in this case, more than 20% believing that playing the lottery is the best wealth accumulation strategy. So, \n\n\(H_0: p = 0.20\)\n\(H_A: p > 0.20\) \n\nWhere \(p\) is the proportion of adult Americans who believe that playing the lottery is the best strategy for accumulating $200,000 in net wealth.

Determine the Level of Significance

The level of significance or alpha (\(\alpha\)), which is the probability of rejecting the null hypothesis when it is actually true, is given as 0.05.

Comparing P-Value with Level of Significance

In hypothesis testing, we reject the null hypothesis \(H_0\) if the p-value is less than or equal to the level of significance \(\alpha\). In this case, we compare the p-value (0.215) with \(\alpha\) (0.05) and since 0.215 > 0.05, we fail to reject the null hypothesis.