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I am currently studying statistics on MIT open course. And I have a question about Z test. Below is the example of IQ test.

H0 = MIT student IQs are distributed identically to the general population = MIT IQ’s follow a N(100, 15^2) distribution.

HA = MIT student IQs tend to be higher than those of the general population = the average MIT student IQ is greater than 100.

And significance level is 0.05

I was wondering if mean and standard deviation of population are given, why can't you just directly calculate the probability of 112 from the population distribution? I can simply type in given information in R. pnorm(x=112, mean=100, sd=225) gives me probability density of IQ 112 or below on the normal distribution with mean 100 and sd 225. So why is the answer not simply 1-pnorm(x=112, mean=100, sd=225)?

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  • $\begingroup$ So you propose to answer the question about MIT students without actually looking at the data altogether? Your question is not about the properties of random variables (probability theory), you should rather be trying to figure out whether the real world agrees with your hypotheses (statistics). $\endgroup$ – tonytonov Aug 3 '17 at 8:53
  • $\begingroup$ Where in your approach do you consider the sample size for MIT students? BTW, I expect MIT students to have a average IQ higher than 112. $\endgroup$ – Joel W. Aug 3 '17 at 13:21
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If you have two populations, then yes, it is that simple. If you have the IQ of every MIT student ever in existence, and the IQ of every person ever in existence, then you can do a simple calculation. Your critical value will be the mean of the general population, since you have every single data point and therefore have 100% confidence in your final value.

The above can't be the case for your question, since they've given you a significance level, and wouldn't model such an easy question as a hypothesis test anyway. I'd imagine that what they're testing is a sample of MIT students against a variable for the 'general population'-population.

In this case, your answer is not a value. A hypothesis test's answer is a statement such as "sufficient evidence at the 0.05 significance level to reject H0". What you need to do is find the critical region (can't calculate it without the sample size, but I assume it's at 112 and above?), find the test value for the MIT sample (e.g 115) and evaluate if it falls into the critical region or not.

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