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.