# Z-Test

**Published by:** [qwedc](https://paragraph.com/@qwedc/)
**Published on:** 2022-08-07
**URL:** https://paragraph.com/@qwedc/z-test-2

## Content

What Is a Z-Test?A z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large. The test statistic is assumed to have a normal distribution, and nuisance parameters such as standard deviation should be known in order for an accurate z-test to be performed.KEY TAKEAWAYSA z-test is a statistical test to determine whether two population means are different when the variances are known and the sample size is large.A z-test is a hypothesis test in which the z-statistic follows a normal distribution. A z-statistic, or z-score, is a number representing the result from the z-test.Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size.Z-tests assume the standard deviation is known, while t-tests assume it is unknown.Understanding Z-TestsThe z-test is also a hypothesis test in which the z-statistic follows a normal distribution. The z-test is best used for greater-than-30 samples because, under the central limit theorem, as the number of samples gets larger, the samples are considered to be approximately normally distributed. When conducting a z-test, the null and alternative hypotheses, alpha and z-score should be stated. Next, the test statistic should be calculated, and the results and conclusion stated. A z-statistic, or z-score, is a number representing how many standard deviations above or below the mean population a score derived from a z-test is. Examples of tests that can be conducted as z-tests include a one-sample location test, a two-sample location test, a paired difference test, and a maximum likelihood estimate. Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size. Also, t-tests assume the standard deviation is unknown, while z-tests assume it is known. If the standard deviation of the population is unknown, the assumption of the sample variance equaling the population variance is made.One-Sample Z-Test ExampleAssume an investor wishes to test whether the average daily return of a stock is greater than 3%. A simple random sample of 50 returns is calculated and has an average of 2%. Assume the standard deviation of the returns is 2.5%. Therefore, the null hypothesis is when the average, or mean, is equal to 3%. Conversely, the alternative hypothesis is whether the mean return is greater or less than 3%. Assume an alpha of 0.05% is selected with a two-tailed test. Consequently, there is 0.025% of the samples in each tail, and the alpha has a critical value of 1.96 or -1.96. If the value of z is greater than 1.96 or less than -1.96, the null hypothesis is rejected. The value for z is calculated by subtracting the value of the average daily return selected for the test, or 1% in this case, from the observed average of the samples. Next, divide the resulting value by the standard deviation divided by the square root of the number of observed values.

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