A simple random sample of 500 students at a certain college were surveyed and asked whether they were satisfied with college life. Two hundred eighty of them replied that they were satisfied. The Dean of Students claims that more than half of the students at the college are satisfied. To test this claim, a test of the hypotheses H0: p = 0.5 versus H1: p > 0.5 is performed.

a. Show that the P-value is 0.004.

b. The P-value is very small, so H0 is rejected. Someone claims that because P is very small, the population proportion p must be much greater than 0.5. Is this a correct interpretation of the P-value?

c. Someone else claims that because the P-value is very small, we can be fairly certain that the population proportion p is greater than 0.5, but we cannot be certain that it is a lot greater. Is this a correct interpretation of the P-value?