A researcher reported that $71.8 \%$ of all email sent in a recent month was spam. A system manager at a large corporation believes that the percentage at his company may be $78 \%$. He examines a random sample of 500 emails received at an email server, and finds that $\mathbf{3 7 0}$ of the messages are spam. Can you conclude that the percentage of emails that are spam differs from $78 \%$ ? Use both $\mathrm{a}=0.01$ and $\alpha=0.05$ levels of significance and the $P$-value method with the table. State the appropriate null and alternate hypotheses.
Solution
Step 1 ：State the null hypothesis (H0) and alternate hypothesis (H1). The null hypothesis is that the percentage of spam emails is 78%, as the system manager believes. The alternate hypothesis is that the percentage of spam emails is not 78%. So, H0: p = 0.78 and H1: p ≠ 0.78, where p is the proportion of spam emails.
Step 2 ：Calculate the test statistic and the p-value. The test statistic in this case is a z-score, which measures how many standard deviations an element is from the mean. The p-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
Step 3 ：The calculated z-score is approximately -2.16, which means the observed proportion of spam emails is about 2.16 standard deviations below the proportion under the null hypothesis. The p-value is approximately 0.031, which is the probability of observing a proportion of spam emails as extreme as 0.74 (or more extreme), assuming the true proportion is 0.78.
Step 4 ：Compare the p-value with the significance levels (0.01 and 0.05) to make a decision about the null hypothesis. If the p-value is less than or equal to the significance level, we reject the null hypothesis. If the p-value is greater than the significance level, we fail to reject the null hypothesis.
Step 5 ：At the 0.01 level of significance, the p-value (0.031) is greater than the significance level, so we fail to reject the null hypothesis. We cannot conclude that the percentage of emails that are spam differs from 78%.
Step 6 ：At the 0.05 level of significance, the p-value (0.031) is less than the significance level, so we reject the null hypothesis. We can conclude that the percentage of emails that are spam differs from 78%.
Step 7 ：So, the conclusion depends on the level of significance. At the 0.01 level, we cannot conclude that the percentage of spam emails differs from 78%, but at the 0.05 level, we can conclude that it does differ.
Step 8 ：Final Answer: \(\boxed{\text{At the 0.01 level of significance, we cannot conclude that the percentage of emails that are spam differs from 78%. At the 0.05 level of significance, we can conclude that the percentage of emails that are spam differs from 78%.}}\)
