A type II error in hypothesis testing describes the event of accepting a null hypothesis that is actually false.
You have made a type II error. Which statement is correct
1The alternative hypothesis is false, and you rejected the null
2.The alternative hypothesis is true, and you rejected the null
3.The alternative hypothesis is false, and you did not reject the null
4.The alternative hypothesis is true, and you did not reject the null
5.None of these are correct
Of the options 1-4, option 4 seems the most correct, logically speaking.
I was always taught not to use the terminology "The alternative hypothesis is true", its best just to say their is evidence to suggest the null hypothesis should be rejected, and you failed to reject it. So on terminology alone I would pick option 5.
It is 4, and terminology isn't an issue IMO. Assuming the model is correct, the alternative is either true or false; a type II error is failing to reject the null hypothesis when the alternative is true.
What people typically have a problem with, terminology wise, is saying "I accept the null hypothesis" or similar statements. In a decision-theoretic context the distinction is meaningless, so I tend not to split hairs, although some instructors (typically introductory-level stat teachers) can be incredibly anal about these things.