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I was given the following exercise:

The Results obtained by 10 students in a test are the following:

72 95 79 83 93 80 91 74 70 86

Test the hypothesis that the mean score is 75.Use two test:one parametric and another non-parametric.

My doubt is for the parametric test, should I make a t-student test? Because I don't have a big sample to use the normal distribution...

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    $\begingroup$ You may find my answer here: Problem understanding what type of test to use and how to proceed to be helpful in thinking about this. $\endgroup$ – gung - Reinstate Monica♦ Nov 12 '12 at 19:58
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    $\begingroup$ Given that this is a homework/textbook exercise, one can neglect the fact that the data are discrete. In order to check the normallity assumption you can use the Shapiro test. In R: x = c(72, 95, 79, 83, 93, 80, 91, 74, 70, 86); shapiro.test(x);. The p-value is $0.65$, then it is reasonable to use a t-test. Final hint: t.test(x, mu=75, alternative="two.sided"). $\endgroup$ – user10525 Nov 12 '12 at 20:00
  • $\begingroup$ What is the level of your class? What tests (both parametric and nonparametric) have you learned? My guess is that your teacher wants to to have an "a ha!" moment when you compare the results of the 2 tests. This could come easily from comparing the results of a t-test and a sign test (or other tests that you have covered), the comparison may then lead you to investigate why they are the same or different (looking at if the usual conditions for the tests hold). You will probably need to find out from your teacher if I have guessed correctly and how strict they are on what tests you use. Note a $\endgroup$ – Greg Snow Nov 12 '12 at 20:17
  • $\begingroup$ yes, i was thinking about t-test and sign test, but isn't sign test about median? $\endgroup$ – user16775 Nov 12 '12 at 20:20
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    $\begingroup$ @Procrastinator, why is there any need to use distribution testing for normality in order to justify using the t-test? That is not one of the usual assumptions for applying a t-test. (Approximate normality of the distribution of the sample mean is the usual assumption.) $\endgroup$ – whuber♦ Nov 12 '12 at 20:25
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Student's t-test is specifically designed to deal with small samples. If your textbook doesn't make that clear, then try another textbook ;-).

For that test to be appropriate, the data need to be plausibly approximately normally distributed. Student's t-test is commonly regarded as being fairly 'robust' in that moderate departures from normality do not muck it up too much. In this specific case you should consider the fact that the distribution of marks may be bounded (e.g. 0% and 100%?) and so cannot be normal. However, if the scores are well away from the bounds then it might be close enough, at least by being roughly symmetrical. (You can explore the approximate distribution by bootstrapping, if you are interested.)

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