Samples that will result in a large p-value

Question

Consider a one-sample t-test regarding the population mean time to complete a task with a one-sided alternative hypothesis of Ha: μ < 10 minutes. A random sample of times to complete the task will be obtained as part of a study. Which of the following values of the sample mean times would result in a p-value of more than 0.5?

1. 7 minutes
2. 9 minutes
3. 11 minutes
4. 13 minutes

I chose 3 and 4 since their values will skew the data towards rejecting the alternative hypothesis. Is this correct? Also, would it be correct to say that "the p-value is the probability that the null hypothesis is true?" (as to interpret it in some vague sense?)

Answer 1

"will skew the data towards rejecting the alternative hypothesis"

-- you don't 'reject the alternative'. You either reject the null or you fail to do so.

would it be correct to say that "the p-value is the probability that the null hypothesis is true?"

No. It's the probability that a sample result at least as extreme as the one observed will occur, given that $H_0$ is true.

Answer 2

3 and 4 is correct for the reasons you mentioned.

However, your interpretation of the p-value is not correct. It is very loosely linked to the probability of H0.

See the wikipedia article on p-value.

In statistical hypothesis testing the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

And from Hubbard, R.; Lindsay, R. M. (article in citations on the wiki):

Using a Bayesian significance test for a normal mean, James Berger and Thomas Sellke (1987, pp. 112–113) showed that for p values of .05, .01, and .001, respectively, the posterior probabilities of the null, Pr(H0 | x), for n = 50 are .52, .22, and .034. For n = 100 the corresponding figures are .60, .27, and .045. Clearly these discrepancies between p and Pr(H0 | x) are pronounced, and cast serious doubt on the use of p values as reasonable measures of evidence. In fact, Berger and Sellke (1987) demonstrated that data yielding a p value of .05 in testing a normal mean nevertheless resulted in a posterior probability of the null hypothesis of at least .30 for any objective (symmetric priors with equal prior weight given to H0 and HA ) prior distribution.