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: 
> 
> "*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*.
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.
> 
> 
>