What exactly is a Z-test?
A z-test is a statistical test to determine whether two population means differ when the large sample size and the variances are known. A normal distribution is assumed for the test statistic, and nuisance parameters such as the standard deviation must be known to conduct an accurate z-test.
Comprehending Z-Tests
The z-test is also a test for the null hypothesis that the z-statistic follows a normal distribution. The z-test is most appropriate for sample sizes greater than 30 because, according to the central limit theorem, the samples are assumed to be approximately normally distributed as the sample size increases.
The null and alternative hypotheses, alpha, and z-score must be stated when conducting a z-test. The results and conclusion should be stated following the calculation of the test statistic. A z-statistic, also known as a z-score, is a number that indicates how many standard deviations are above or below the mean population of a z-test score.
Tests such as the one-sample location test, the two-sample location test, the paired difference test, and the maximum likelihood estimate can be conducted as z-tests. Z-tests are closely related to t-tests, but t-tests perform better when the sample size of an experiment is limited. Additionally, t-tests presume that the standard deviation is unknown, whereas z-tests assume it is known. If the population’s standard deviation is unknown, it is assumed that the sample variance equals the population variance.
One-Sample Z-Test Illustration
Assume an investor wants to determine if the average daily return of a stock is greater than 3 percent. The average of a straightforward random sample of 50 returns is calculated to be 2%. Assume that the return’s standard deviation is 2.5%. Therefore, the null hypothesis holds when the mean, or average, is 3%.
In contrast, the alternative hypothesis is whether the average return is greater than or less than 3 percent. Consider a two-tailed test in which an alpha of 0.05% is chosen. As a result, 0.025% of the samples are in each tail, and the alpha has a critical value of either 1.96 or -1.96. The null hypothesis is rejected if its value exceeds 1.96 or falls below -1.96.
The value of z is determined by subtracting the value of the average daily return chosen for the test, which in this instance is 1%, from the observed sample mean. After that, divide the resultant value by the square root of the observed values.
The test statistic is therefore: (0.02 – 0.01) / (0.025 √50) = 2.83
Since z is more significant than 1.96, the investor rejects the null hypothesis and concludes that the average daily return is greater than 1%.
How Does a T-Test Differ from a Z-Test?
Z-tests are closely related to t-tests, but t-tests are most appropriate when the sample size is modest, less than 30. Additionally, t-tests presume that the standard deviation is unknown, whereas z-tests assume it is known.
When Should a Z-Test Be Used?
Suppose the population standard deviation is unknown and the sample size is greater than or equal to 30. In that case, the z-test should be used to assume that the sample variance equals the population variance. If the population standard deviation for a variable is unknown, regardless of the sample size, a t-test should be used instead.
What exactly is a Z-score?
A z-score, also known as a z-statistic, represents the number of standard deviations above or below the mean population that a z-test score is. It is a numerical measurement that describes the relationship between a value and the mean of a group of values. If a z-score is 0, it indicates that the score of the data point is the same as the mean score. One standard deviation from the mean would correspond to a z-score of 1.0. Z-scores may be positive or negative, with a positive value denoting that the score is above the mean and a negative value denoting that the score is below the mean.
Central Limit Theorem (CLT): What Is It?
In probability theory, the central limit theorem (CLT) states that as sample size increases, the sample distribution approaches a normal distribution (also known as a “bell curve”), assuming that all samples are the same size regardless of the population distribution shape. For the CLT to accurately predict the characteristics of a population, sample sizes equal to or greater than 30 are deemed sufficient. The CLT holding is reliant on the fidelity of the z-test.
The conclusion
- The z-test is a statistical test to determine whether the means of two populations differ when the variances are known and the sample size is large.
- A z-test is a hypothesis test where the z-statistic has a normal distribution.
- A z-statistic, also known as a z-score, is a number that represents the outcome of the z-test.
- Z-tests are closely related to t-tests, but t-tests perform better when the sample size of an experiment is modest.
- The Z-test assumes that the standard deviation is known, whereas the T-test assumes that it is unknown.
