What Is a Z-Test?

A z-test may be a statistical test won’t to determine whether two population means are different when the variances are known and therefore the sample size is large. The test statistic is assumed to possess a traditional distribution, and nuisance parameters like variance should be known so as for an accurate z-test to be performed.

A z-statistic, or z-score, may be a number representing what percentage standard deviations above or below the mean population a score derived from a z-test is.

KEY TAKEAWAYS

A z-test may be a statistical test to work out whether two population means are different when the variances are known and therefore the sample size is large.

It is often wont to test hypotheses during which the z-test follows a traditional distribution.

A z-statistic, or z-score, may be a number representing the result from the z-test.

Z-tests are closely associated with t-tests, but t-tests are best performed when an experiment features a small sample size.

Also, t-tests assume the quality deviation is unknown, while z-tests assume it’s known.

How Z-Tests Work

Examples of tests which will 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 associated with t-tests, but t-tests are best performed when an experiment features a small sample size. Also, t-tests assume the quality deviation is unknown, while z-tests assume it’s known. If the quality deviation of the population is unknown, the idea of the sample variance equaling the population variance is formed.

Hypothesis Test

The z-test is additionally a hypothesis test during which the z-statistic follows a traditional distribution. The z-test is best used for greater-than-30 samples because, under the central limit theorem, because 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 therefore the results and conclusion stated.

One-Sample Z-Test Example

Assume an investor wishes to check whether the typical daily return of a stock is bigger than 1%. An easy random sample of fifty returns is calculated and has a mean of twenty-two. Assume the quality deviation of the returns is 2.5%. Therefore, the null hypothesis is when the typical, or mean, is adequate to 3%.

Conversely, the choice hypothesis is whether or not the mean return is bigger than 3%. Assume an alpha of 0.05% is chosen with a two-tailed test. Consequently, there are 0.025% of the samples in each tail, and therefore the alpha features a critical value of 1.96 or -1.96. If the worth of z is bigger than 1.96 or but -1.96, the null hypothesis is rejected.

The value for z is calculated by subtracting the worth of the typical daily return selected for the test, or 1% during this case, from the observed average of the samples. Next, divide the resulting value by the quality deviation divided by the root of the amount of observed values. Therefore, the test statistic is calculated to be 2.83, or (0.02 – 0.01) / (0.025 / (50)^(1/2)). The investor rejects the null hypothesis since z is bigger than 1.96 and concludes that the typical daily return is bigger than 1%.