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I'm a student in AP Statistics and our end-of-year project has me stumped on one small aspect.

The variables here aren't the ones in my project, but it'll make it easier to explain.


I'm trying to determine whether the average male IQ is different from the average female IQ. I took $x$ males and determined their average IQ to be $m_x$ with sample standard deviation $s_x$. I took $y$ females and determined their average IQ to be $m_y$ with sample standard deviation $s_y$.

I then did the following procedures:

  1. I conducted a two-sample $t$-test on my data and concluded at a significance level of $\alpha = 0.05$ that the average male IQ and average female IQ are not significantly different.

  2. I computed $95$% confidence intervals for $m_x$ and $m_y$ and found that:

    • $m_y$ is within the confidence interval for $m_x$
    • $m_x$ is not within the confidence interval of $m_y$

These procedures seem to produce conflicting results.

  • My two-sample $t$-test indicates that $\mu_x = \mu_y$.
  • My confidence interval for $m_x$ indicates that $\mu_x = \mu_y$.
  • My confidence interval for $m_y$ indicates that $\mu_x \ne \mu_y$.

At the end of the day, I'm probably going to use the two-sample $t$-test to substantiate my claim, but why am I getting different results?

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    $\begingroup$ Could you give us a reference to the procedure that compares a sample mean to a confidence interval? $\endgroup$ – whuber Apr 21 '16 at 20:14
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    $\begingroup$ Heads up: to avoid confusing your statistics, do not use $\mu_x$ and $m_x$ interchangeably. The sample mean and population mean are different. $\endgroup$ – daOnlyBG Apr 21 '16 at 20:14
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    $\begingroup$ Slightly off-topic, but hilarious - my experience on an online practice AP Statistics test or-exchange.org/questions/12345/… "the number of presidential elections in the 20th century is an integer, but it does not vary and it does not result from a random process; so it is not a random variable." - OMG, better throw out all my probability books. $\endgroup$ – Mark L. Stone Apr 21 '16 at 20:32
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Yes, you should use the two-sample t-test. Here is an example you can use the coverage of the confidence interval to determine the significance. Suppose you want to test if the IQ of man is different from 100, you can construct the confidence interval of male IQ and check if 100 follows in the interval. Here 100 is a CONSTANT, not a random statistic. If you want to use confidence intervals to determine the significance, you must check if there is any overlap between the two confidence intervals. If there is some overlap, they are not significantly different.

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