Maths question check

Hi please could I have help on this question? So I’ve done the question and I think I’ve done the same thing as the ms? The only thing is I got n=15.3664… and then I put in the values n=15 and n=16 into the calculator to see which gave the correct value. The ms has used an inequality but I’m unsure what they mean so I was hoping to get some help on that part based on my working?
Thank you.

It should be ok as the last column in the ms says you can work with = rather than an inequality, though it may be worth a brief mention that youre working with a critical value / equality, then reason about the inequality at the end.
One thing though, question part c) talks about the refined model which seems to be the model with the refined mean (17) in question part b). In this case, the only difference with using 15 is the sign of x-mu and that gets absorbed with the +/-1.96. The two inequalities on the 2nd line of the ms make sense if mu=17. Maybe you should be a bit clearer about this step in your working where you go from two inequalities to a single equality.

sorry, how did you work out the refined mean, i'm not sure i've come across it?
thanks!

Question part ii) talks about the "refined model" with a mean of 17 rather than 15. Obv you work out the 17 in ii) and its the same distance from 16. But you mention 15 in your working which is incorrect for iii).
Id have done a sketch of a bell curve for the mean, centered on 17 and the fact youre testing 16 so
https://b28mathstutor.co.uk/hypothesis-testing-for-normal-a-level-maths/
and write
xbar ~ N(17, 4/n)
Then youre testing xbar=16 in the left tail at 0.025 and its pretty much as youve done/the ms. But its clearer about signs and neglecting the right tail
(xbar - 17)/ 2/sqrt(n) = -1.96
...

Ohhh i get what you mean now, that makes a lot more sense!
just one other question, if it were greater than instead of less than, would my working be correct and what would the inequality sign be?
thanks!

Youve done the right tail with a mean of 15. So the "right" working for the "wrong" question. However you dont really explain what youre doing clearly, so a sketch of the distribution with the mean, tails, ... can make it clearer for both you and the marker.