Step 1 :Define the null and alternate hypotheses as follows: \(H_0: p = 0.69\) (The percentage of spam emails is 69%) and \(H_1: p > 0.69\) (The percentage of spam emails is greater than 69%).
Step 2 :Identify that this is a right-tailed test.
Step 3 :Calculate the sample proportion \(\hat{p}\) as \(\hat{p} = \frac{365}{500} = 0.73\).
Step 4 :Calculate the test statistic z using the formula: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 * (1 - p_0)}{n}}}\).
Step 5 :Substitute the values into the formula to get: \(z = \frac{0.73 - 0.69}{\sqrt{\frac{0.69 * (1 - 0.69)}{500}}} = 1.93\).
Step 6 :The P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. It can be found using a standard normal distribution table or a calculator with statistical functions.
Step 7 :For a right-tailed test, the P-value is the area to the right of the test statistic.
Step 8 :Given the z-score of 1.93, the P-value would be less than 0.05, indicating strong evidence against the null hypothesis at the \(\alpha = 0.05\) level of significance.
Step 9 :For the \(\alpha = 0.01\) level of significance, we would need to compare the P-value to 0.01 to make a conclusion.
Step 10 :Unfortunately, without a standard normal distribution table or a calculator with statistical functions, we can't calculate the exact P-value.