Dear all,
It is mentioned in the Stats books (both of local and international authors) that in hypothesis testing we use z-test when pop SD is known and use t-test when pop SD is unknown (method 1). I have come across the case of large sample sizes (n > 30) in some Stats even of Research books (both of local and international authors) that for large sample sizes (n > 30) we should use still z-test but by replacing the pop SD by sample SD value in formula (method 2). Kindly, suggest that which method is correct and which one is currently being used by the Stats practitioners and Gurus or in the modern Stats?
Hello Syed,
The advice that (a) N > 30 is a "large" sample; and (b) when one has a large sample, ignore the sample degrees of freedom and treat t as z stems from
an era when statistical software didn't exist, and one had to rely on look-up tables of critical values in order to judge statistical significance.
It is true that, as N increases, the associated t-distribution more closely resembles the standard normal distribution. However, where is the fit the worst? In the tails. Where are so-called critical values for significance found? In the tails. So, when does N become large enough? That depends on the statistic of interest and your willingness to bias the test (however slightly).
If you're stranded on an island without phones, or lights, or motor cars (or statistical software), just use +/- 2 as your critical value (for 95% CIs, or for 5% alpha levels). Otherwise, let the statistical software correctly evaluate the probability for you.
Good luck with your work!
Hello Syed,
The advice that (a) N > 30 is a "large" sample; and (b) when one has a large sample, ignore the sample degrees of freedom and treat t as z stems from
an era when statistical software didn't exist, and one had to rely on look-up tables of critical values in order to judge statistical significance.
It is true that, as N increases, the associated t-distribution more closely resembles the standard normal distribution. However, where is the fit the worst? In the tails. Where are so-called critical values for significance found? In the tails. So, when does N become large enough? That depends on the statistic of interest and your willingness to bias the test (however slightly).
If you're stranded on an island without phones, or lights, or motor cars (or statistical software), just use +/- 2 as your critical value (for 95% CIs, or for 5% alpha levels). Otherwise, let the statistical software correctly evaluate the probability for you.
Good luck with your work!
The reply given by MR. David Morse is very right. We know as N increases, the associated t-distribution more closely resembles the standard normal distribution. Further, t-test may be used in case of both small sample ( n30), but Z-test can be used in case of large samples only. But presently, software are used for the purpose and generally we find exact p values in place of 5% or 1% alpha.
I agree with both Morse and Bodla but the theory of using the t test for small samples and the z test for large samples is still valid. It is taught to students in their lectures and should still be taught. This is in spite of the availability of computers and softwares. The computer user should know the theories underlying the applications.
Hello ... this is from Medical statistics 2nd Edition for Kirkwood and Sterne
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Sampling Design
- Sample Size