If the population size is large and we do not either know the population mean and its standard deviation. can we run z-test to compare the mean of a sample to a hypothesized mean? if so, how? please share some book or research paper for reference. Thanks in advance
Muhammad,
In order to run a Z-test to compare the mean of a sample to the hypothesized mean, the population mean and standard deviation are absolutely necessary - you could run your test with a hypothesized population mean and stdv but your statistical inference will most definitely be inaccurate.
However, based on the CLT - given a large enough sample size (not population), you can assume normality of your sample and carry out the test with your hypothesized value. Note that a "large enough" sample size would be case dependent and I would consult field experts in determining whether the sample size is sufficient or not. You can refer to "Statistical Methods - Freund 3rd edition" book for more details. Hope this helps.
If both mean and standard deviation are unknown you should run a t-test rather than a z-test. If the distribution is known to be Gaussian, the t-test is valid even if the sample size is small. If the distribution is unknown a t-test is applicable if the sample size is not very small.
While Peter is correct - once n is large enough then the one sample z test and t test are indistinguishable. For example with n = 500 the critical value for a two-tailed test is identical to 2 d.p.
> qt(.975,500)
[1] 1.96472
> qnorm(.975)
[1] 1.959964
For many practical purposes n = 30 might be sufficiently accurate. With other t tests it will depend on n per group and the variability in each group and becomes riskier to use a z approximation.
there seems to be some confusion about the nature of statistical testing...
suppose a baker scales10 loafs of bread to ensure that the mean weight of his loafs is sufficiently close to an ideal weight of 1 kg. by doing so he compares the mean weight from his sample with the hypothetical weight of 1 kg. this value of 1 kg is the null-hypothesis of his test, and as the name says, it's hypothetical! in this sense, the population mean is "always" hypothetical. in other words, if the true mean of "all loafs he ever bakes" were known then there would be no need for statistical inference!
in order to make sure that the difference of his sample mean and the ideal weight is not too big, the baker needs information about the variability of his loafs' weights. since his sample comes from the whole population of all loafs of bread, he can use the population variance, if it is known from somewhere... in this case, he performs a z-test. if on the other hand the population variance is not known then he just uses the "next best thing", i.e. the sample variance, as an approximation for the unknown population variance. in this latter case, the baker performs a t-test.
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