Final answer:
To answer whether there is evidence to believe the magazine's claim about the proportion of adults buying takeout food daily, we conduct a hypothesis test comparing the sample proportion to the claimed proportion. The test will involve setting null and alternative hypotheses, determining the significance levels, calculating the p-value, and making a decision based on the p-value and the predetermined levels of significance (α). The claim will be accepted or rejected based on the outcome of this hypothesis testing procedure.
Step-by-step explanation:
The problem presented is one of hypothesis testing in statistics, specifically testing the proportion of adults buying takeout food daily against a magazine's report. To determine if there is evidence to support or reject the magazine's claim that 11% of adults buy takeout food every day, we establish the null hypothesis (H0) and the alternative hypothesis (H1). In this case, the null hypothesis will state that the true proportion of adults buying takeout food daily is 11% (H0: p = 0.11), and the alternative hypothesis will contend that the true proportion is not equal to 11% (H1: p ≠ 0.11).
To carry out the hypothesis test, a significance level (α) needs to be set. The significance level is the probability of rejecting the null hypothesis when it is actually true. The provided data includes two levels to consider, α = 0.02 and α = 0.05. Using these levels, we would reject the null hypothesis if the calculated p-value from the survey data (32 customers out of 200 buying takeout food daily) is less than α.
The next step involves calculating the p-value for the test statistics which is calculated using the sample proportion and comparing it to the assumed population proportion under the null hypothesis. If this p-value is lower than the chosen significance level, we reject the null hypothesis, indicating there is evidence against the claim. If the p-value is higher, we fail to reject the null hypothesis, implying the sample data does not provide strong evidence against the magazine's claim.
Once the data is analyzed, we reach a conclusion on whether the magazine's claim can be accepted or rejected based on the p-value and the predetermined levels of significance.