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We know from several population studies that the mean concentration of protein in the blood of healthy people is 2.5. We are given a data set (of patients) of $n=10$ and we obtain the mean=3.1 and standard deviation=1.1.

I want to use a t-test to test if the concentration of the protein in the given data set is significantly higher than that in the population.

The null hypothesis would be that $\mu \leq 2.5$ and the alternative hypothesis would be that $\mu >2.25$.

And then we calculate the $t$ value, we also find the critical value for significance level 0.05. Which one of the two hypotheses we can reject if we get a $t$ value less than the critical value?

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  • $\begingroup$ Seems a self study question, if so please add the [self-study] tag. $\endgroup$
    – utobi
    Commented Nov 26, 2016 at 20:09

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If you replace 2.25 with 2.5 in the above statement the hypothesis test will be set up correctly.

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  • $\begingroup$ Yes you are right. So what about rejecting the hypothesis? Do we reject the null or the alternative one? $\endgroup$
    – ChFor D
    Commented Nov 26, 2016 at 18:42
  • $\begingroup$ @ChForD: With frequentist hypothesis testing, you either reject the null hypothesis or you do not reject the null hypothesis at the specified significance level. If you reject the null hypothesis, all you have left is the alternative hypothesis. If you do not reject the null hypothesis, you still have the null hypothesis, and can make no comment about the alternative hypothesis (which you have not tested). $\endgroup$
    – Henry
    Commented Nov 26, 2016 at 19:00

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