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A group of 50 complete a national fitness test and get a mean score of 80 out of 100. The national average is 72 with standard deviation 6. Can we conclude the group of 50 is fitter than the national average. Let ${\mu}$ be the national average score. What are the null and alternative hypothesis?

I'm really confused on this on. Its almost as if I should be using a difference of means test.

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The null hypothesis is $$H_0: μ=72$$ and the alternative hypothesis is $$H_0: μ>72$$ If you reject the null hypothesis then this means that your sample of $50$ comes from a "different population" i.e. from a population with higher fitness score. You do not need a difference of means test. Just take a mean test, you can use the $z$ statistic since $n=50>30$.

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  • $\begingroup$ So ifs almost like a proof by contradiction. We know the mean is 72, so if I end rejecting H_0, that means the sample came from a 'different population'? $\endgroup$ – user108605 Nov 14 '14 at 17:43
  • $\begingroup$ Before I answer, can you tell me how familiar you are with hypothesis testing, so that I can fit my answer? $\endgroup$ – Jimmy R. Nov 14 '14 at 17:45
  • $\begingroup$ fairly familiar - I know all the terminology $\endgroup$ – user108605 Nov 14 '14 at 17:46
  • $\begingroup$ :) Ok. So roughly: With a hypothesis test you can answer the question" assuming the null hypothesis is true, how likely is it that I will take a sample like I did" (in your case with mean 80) If it is very unlikely then the result of the test will be to reject the null, if not to accept it. Chances are always for null hypothesis. So, what bothers me in your question, is that you want to draw a conclusion for the sample of the 50 and not for the population behind it. That is unusual as also the other posted answer explains. $\endgroup$ – Jimmy R. Nov 14 '14 at 17:52
  • $\begingroup$ yes I think its quite slipshod question $\endgroup$ – user108605 Nov 14 '14 at 17:57
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As stated, the problem doesn't make a lot of sense as a hypothesis test. That this group of 50 scored higher than average is an empirical observation, so there's no occasion for statistical inference.

But if the question were whether the group is significantly different from the national average, then it would make sense, because the conclusion that rejects the null hypothesis would not be that the group scored above average, but rather that the population of which the group is a sample is fitter than average. In that case, the null hypothesis would be that the population from which the group is a sample is not fitter than the nation as a whole.

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