Question

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?

Short answer

There isn't convincing evidence, based on this sample, to support the claim that more than 20% of adult Americans believe that playing the lottery is the best strategy for accumulating $200,000 in net wealth. Our p-value of 0.2148 is greater than the common significance level of 0.05, so we do not reject the null hypothesis of p <= 0.20.

Step-by-step solution

Defining the Null and Alternate Hypothesis

Let's first define the Null Hypothesis (H0) and the Alternate Hypothesis (Ha). The null hypothesis is that 20% or less of American adults believe the lottery is the best way to accumulate $200,000, i.e., H0: p <= 0.20. The alternate hypothesis is that more than 20% believe that the lottery is the best approach, i.e., Ha: p > 0.20.

Calculate the Test Statistic

Now, let's calculate the test statistic. The formula for the test statistic in a proportion hypothesis test is z = (p-hat - p0)/sqrt[(p0(1-p0))/n] where p-hat is the sample proportion, p0 is the population proportion under the null hypothesis, and n is the sample size. Here, p-hat = 210/1000 = 0.21, p0 = 0.20, and n = 1000. Substituting these values into the formula gives z = (0.21-0.20)/sqrt[(0.20(1-0.20))/1000] = 0.01/sqrt[0.00016] = 0.01/0.01265 = 0.7905.

Compute the P-Value

From the standard normal (Z) distribution table, we can find the probability that Z is more than 0.7905 (the value of our test statistic). However, we want the probability that Z is greater than this, which is 1 - the value we just found. This gives us a p-value of 1 - 0.7852 = 0.2148.

Decision

If the p-value is less than the significance level (generally 0.05), the null hypothesis is rejected. However, here the p-value is 0.2148 which is greater than 0.05. Thus, the null hypothesis cannot be rejected. This means there isn't strong evidence to claim that more than 20% of adult Americans believe that playing the lottery is the best strategy for accumulating $200,000 in net wealth.