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I'm having a problem with this question:

Record shows that the students in a certain university has mean IQ of 115 with a standard deviation of 10. A research conducted on 30 students showed a mean IQ of 110. What is the probability that the sample of 30 students will have a mean IQ less than 115?

Our class came up with 3 different answers:

A.) .9969
They computed this using 110 as the population mean. [(115-110) / (10/√30)] then z-table.

B.) .0031
They computed this using 115 as the population mean then 110 as the random variable. [(110-115) / (10/√30)] then z-table.

C.) .5000
They computed this using 115 as the population mean then 115 as the random variable. [(115-115) / (10/√30)] then z-table.

In the end, our class accepted A as the answer but I'm not convinced. What are your thoughts about this?

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    $\begingroup$ The question you were asked is not as clear as it might be, as it depends on what "the sample of 30 students" refers to. If it refers to the sample already taken, the probability is 100%, since that sample actually had a mean IQ of $110 < 115$. If it's a whole new sample, it's C, as the population from which the new sample is drawn has a mean of 115, so that's the appropriate population mean to use. In this latter case, the 110 has nothing to do with it, because it refers to the old sample, which is different to the new one, so it's not clear why it's here except to confuse. $\endgroup$
    – jbowman
    Commented Feb 17, 2018 at 16:38
  • $\begingroup$ If the answer is not 100% (for the trivial and uninteresting interpretation of the question), then it must depend on the distribution of IQs among the students. For instance, the university could have 9900 students with IQs of 114 each and 100 students with IQs of 214: the chance that a random sample of 30 has a mean IQ under 115 would then be 74%. $\endgroup$
    – whuber ♦
    Commented Feb 18, 2018 at 0:04

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I believe jbowman's first reading is correct. The wording is "the sample of 30 students", not "a (new) sample of 30 students". Hence, the sample mean is known to be 110, which is less than 115. The probability that this sample mean is below 115, if you define it at all, can only be 1.

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    $\begingroup$ One could interpret the research results as being new measurements of IQ, subject to error, and therefore (to assess their plausibility) ask the question anyway: what is the chance that the average of the university's official records of IQs of the sampled students is less than 115? That is a meaningful--and far more interesting--question. $\endgroup$
    – whuber ♦
    Commented Feb 18, 2018 at 0:07

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