Is there any evidence that he average differs from 45

I've worked out the mean to be 39.53

In that case, there obviously is evidence that the mean differs from 45. You've just produced it. The question is whether that evidence is statistically significant - you're going to need to do a hypothesis test, testing at some significance level which you'll have to pick before you start.

So would my null hypothesis be mean = 39.53
and alternative would be mean is greater than 39.95

and use t-distribution and reject the null is p(X>39.95)<1%?

No - imagine you said, "I want to test whether the mean time is 45. What test will I perform so that I can just plug my data in right at the very end?"
Your null hypothesis is that the mean is 45. The data will lead you to reject that mean=45 or be insufficient to reject that mean=45.
If instead you used the null hypothesis that the mean is 39.53, and your data led you to reject that mean=39.53, you have learnt nothing about the original problem of whether the mean is 45.

Also, you need to know what underlying distribution the data came from. You suggest that it's a t-distribution, but you didn't write it in the OP. Hypothesis tests tell you about the parameters of a distribution, not what the distribution was in the first place.

Ok that makes sense now and the question hasn't specified the type of distribution. How would I go about determining the that. The observations are profit of 20 stores. That's all I'm given with their level of sales.

Hmm - in that instance, I'd be inclined to say that "yes, we have the best possible evidence - I calculated the mean and it was different" :P unless you're meant to assume that profits are normally distributed? Not sure, I'm afraid… is it a single shop, with observations over time? (Otherwise there's no guarantee at all that they'll have the same distribution.)

a shop with 20 different branches.

a shop with 20 different branches.


Also, if thats the case does that mean using t-test for the next part of the question is invalid too since the distribution is unknown.

What's the actual question/situation here?

I can't think of any a priori reason why value of store sales would be expected to have any particular distribution or mean - in the "real world" there are so many underlying factors that affect such a thing.

I have come across situations where a particular "sales initiative" is being trialled to see if it increases average sales for example, but that doesn't seem to be the situation you're describing.

I've posted a picture of the original question.

Hmm....I'm not a statistics specialist and I have to say I'm not sure what they're expecting for this. I would have thought that you would need to make some assumption about the underlying distribution of things like "lines stocked" and "profits" in order to make statements about "significance" but I could be wrong.

Have you come across any similar questions or case studies as part of your course?