Null hypothesis testing and wording

A random sample of size 40 has mean = 120. The p value for testing $H_0$: μ = 100 against $H_1$: μ ≠ 100 is p = .057. Explain what is incorrect about each of the following interpretations of this p value, and provide a proper interpretation.

n=40, $\bar x=120$, $H_0:μ=100$, $H_1:μ≠100$, p-value=0.057

So I have the following statement about the probability, is it true or not true?

statement 1 The probability has mean = 120 if $H_0$ is true equals 0.057

This is what I think: the statement might be accurate but the wording might not be clear. The p-value 0.057 represents the probability of the mean of the population of the subject we want to study equals to 120, given $H_0$ is true. (is 120 the mean of the sample or the mean of the population of the subject we want to study?)

There's another statement that the wording confuses me a lot:

statement 2 If in fact μ ≠ 100, the probability equals 0.057 that the data would be at least as contradictory to $H_0$ as the observed data

What I think: A p-value is the probability of observing a value of the test statistic at least as contradictory to $H_0$ (favoring $H_1$) as the observed value, when $H_0$ is assumed to be true. So, this is incorrect for μ not equaling to 100.

If μ≠100, then first, we would expect a different p-value that is smaller than 0.05. On the other hand, there is no guarantee that the data would be contradicting to $H_0$ as the exact same probability of 0.057.

Is that right?

statement 3 We can accept $H_0$ at the α = 0.05 level.

statement 4 We can reject $H_0$ at the α = 0.05 level

Is one of the above statement true, or should it all be false of the p-value is 0.057. I know that we are supposed to reject the null hypothesis if p<0.05, but since 0.057 is really close to 0.05, does that change anything?

Thank you very much for discussing the issues with me.

• Welcome to our site! Please add the [self-study] tag & read its wiki – Silverfish Sep 28 '16 at 6:14
• You might want to replace the "population" tag by the "p-value" tag which is probably more relevant here. I suggest you have a read of some of our other questions with the hypothesis testing and p-value tags as your question has a lot of overlap with previous ones – Silverfish Sep 28 '16 at 6:16
• I'm going to be pedantic (though that might be the intent of the questions) and point out that you never accept the null hypothesis. You say "can't reject null hypothesis", meaning your experiment is insufficient to determine whether either hypothesis (null or alternate) is true or false. Not sure if that helps. – barrycarter Sep 28 '16 at 13:11