The Z-test is used to compare means of two distributions with known variance. One sample Z-tests are useful when a sample is being compared to a population, such as testing the hypothesis that the distribution of the test statistic follows a normal distribution. Two sample Z-tests are more appropriate for comparing the means of two samples of data.
Requirements for the Z-test:
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The mean and standard deviation of the population distribution are known
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The mean of the sample distribution is known
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The variance of the sample is assumed to be the same as the population
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The population is assumed to be normally distributed
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In cases where the population variance is unknown, or the sample size is less than 30, the Student's t-test
may be more appropriate.
To calculate a Z-test statistic, the following formula can be used:
z = ,
z = ,
where, x is the sample mean, m is the population mean, and SE is the standard error, which can be calculated using the following formula:
SE = ,
where, s is the population standard deviation and n is the sample size.
For each significance level, α, the Z-test has a critical value. For example, the significance level α = 0.01, has a critical value of 2.326. If the Z-test statistic is greater than this critical value, then this may provide evidence for rejecting the null hypothesis.
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