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I have the following hypothesis

  • $\mu$=20
  • $\mu$>20

I am required to generate 1000 samples with sample size(n)=5 from the following 3 populations.

  • N(21, 4)
  • N(22, 4)
  • N(23, 4)

then I calculated the z value for each of these samples using the following formula

Z=(sampmean-20)/(2/sqrt(5))

and counted the number of observations that fall outside the 1.645 cut off.

what I found is that as the as the the population mean increased from 21 to 23 the type 1 error rate (or the observations that fall outside of 1.645) seemed to increase. Is that correct

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  • $\begingroup$ In these three cases, you can only make a Type II error, not a Type I error, as the null hypothesis is clearly false (i.e., the true population from which you are sampling does not have a mean of 20). $\endgroup$
    – dbwilson
    Mar 14, 2018 at 16:44

1 Answer 1

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Seems to me that the mean isn't the important factor, the sample size is. In general, the larger the sample size the less likely you are to make a type 1 error. Stndrd error rates decrease as sample sizes increase. Variability around the mean, not the mean itself, is important.

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  • $\begingroup$ The Type I error rate remains the same as the sample size increases. What changes is how small of an effect you can find to be significant (e.g., you have more statistical power). Stated differently, Type II is affected by sample size but Type I is not. Our p-values would be meaningless if that were not the case as they reflect the likelihood of a Type I error. $\endgroup$
    – dbwilson
    Mar 14, 2018 at 18:29

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