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Z-test calculator


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Z-test
For sample means, the z-score is defined as
z = ( <X> - m ) / ( s / n1/2 ).
For a one-sample z-test with known population mean and standard deviation, the sample size n = 1. With the z-test calculator, users can easily calculate z-score from the p-value or the vice versa. The population standard error s may be replaced with the sample standard error s if the sample size n is sufficiently large.

Example (Traynor et al. Neurology. 1999 Feb;52(3):504-9):
The national average annual incidence rate of ALS is 2.8 per 100,000 population over 15 years of age (95% CI 2.4-3.1). The eastern region has a higher incidence rate of 4.4 per 100,000 person-years. One-tailed z-test can be used to determine whether the eastern region has higher ALS incidence rate.
H0 The null hypothesis: the annual incidence difference m between the national (mn) and the eastern region (me) is 0, i.e., m = mn - me = 0.
Ha The alternative hypothesis: m = mn - me < 0.
We find, z = (y - m) /s = y/s = (yn - ye)/(sn + se) ~ (yn - ye)/(2 * sn) ~ (2.8-4.4)/0.7 ~ -2.3. Thus p = 0.01. The null-hypothesis should be rejected in favor of the alternative hypothesis, i.e., high ALS incidence in the eastern region is statistically significant.

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