Z-Test: Simplified Stock Analysis
Statistical Test Analysis: Examination of Z-Test, Its Applications, and an Illustrative Example
Ever wondered how an investor deciphers the average daily return of a stock? Enter the z-test, a nifty statistical tool that helps sort things out.
Z-Tests: A Closer Look
Z-tests, essentially hypothesis tests, help compare one mean (average) with a predetermined value. They assist in determining whether two means are alike. For the test to work its magic, the data must approximately follow a normal distribution. Some calculations are in order, such as variance and standard deviation, to perform the z-test.
Key Takeaways:
- Z-tests are for data following a normal distribution.
- Z-tests analyze a z-statistic, a number demonstrating the z-test's result.
- Z-tests are closely related to t-tests, but the latter are ideal for smaller sample sizes.
- In contrast to t-tests, z-tests assume the standard deviation is known.
Comprehending Z-Tests
Z-tests also belong to the family of hypothesis tests where a z-statistic follows a normal distribution. When the experiment involves greater-than-30 samples, thanks to the central limit theorem, the samples can be considered approximately normally distributed.
When conducting a z-test, one must state the null and alternative hypotheses, alpha level, and calculate the z-score, commonly called the test statistic. The z-score lies in the realm of how many standard deviations the score derived from a z-test is above or below the mean population.
Z-tests encompass various tests, like the one-sample location test, two-sample location test, paired difference test, and maximum likelihood estimate. Z-tests and t-tests share a kinship, but t-tests excel with smaller sample sizes and bear the assumption that the standard deviation is unknown.
Z-Score Formula
Z = ( x - μ ) / σ
z - Z-scorex - Evaluated valueμ - Meanσ - Standard deviation
One-Sample Z-Test Example
In this scenario, an investor aims to test if the average daily return of a stock exceeds 3%. A simple random sample of 50 returns is calculated, with an average return of 2%. With a known standard deviation of 2.5%, the null hypothesis is the average (mean) is equal to 3%.
The alternative hypothesis questions whether the mean return is above or below 3%. Let's say an alpha of 0.05% is chosen with a two-tailed test. Consequently, there's a 0.025% chance of samples in each tail, and the alpha has a critical value of 1.96 or -1.96. If the z-value surpasses 1.96 or falls below -1.96, the null hypothesis is discarded.
To calculate the test statistic:
- Subtract the average daily return of interest (3% in this example) from the observed average of the samples.
- Divide the resulting value by the standard deviation divided by the square root of the number of observed values.
The investor rejects the null hypothesis as z is less than -1.96 and concludes that the average daily return is less than 3%.
Differences Between T-Test and Z-Test
Z-tests have ties with t-tests, but the latter perform better in data consisting of a small sample size (\<30). T-tests are also used when the standard deviation is unknown, whereas z-tests assume it is known.
When to Use a Z-Test
Z-tests are advisable when the population standard deviation is known, and the sample size is 30 or more data points. If the population standard deviation is unknown, a t-test should be employed instead.
Z-Score and Central Limit Theorem (CLT)
Z-score, or z-statistic, quantifies the relationship between a score derived from a z-test and the mean of a group of values. A z-score of 0 indicates that the data point's score aligns perfectly with the mean score. A z-score of 1.0 signifies a value one standard deviation from the mean. z-scores can be either positive or negative, with a positive value indicating the score exceeds the mean and a negative score suggesting it falls below the mean.
The central limit theorem (CLT), a cornerstone in probability theory, stipulates that a sample distribution, provided all samples are identical in size and regardless of the population distribution shape, approaches a normal distribution (or "bell curve") as the sample size increases. Sample sizes equal to or greater than 30 are considered sufficient for contrasting the characteristics of a population with precision. The accuracy of z-tests relies on the CLT's validity.
Assumptions of the Z-Test
For a z-test to be accurate:
- The population must be normally distributed.
- The samples must exhibit identical variance.
- Data points must be independent of one another.
Z-Test: Wrapping Up
A z-test is a valuable aid in hypothesis testing, particularly when deciding whether two means are the same (the null hypothesis). It works splendidly when the population standard deviation is known and the sample size includes 30 or more data points. In other scenarios, a t-test would be more effective.
- In the realm of financial trading, defi platforms can utilize z-tests to determine the likelihood of a token's average daily return deviating from a specified value, enhancing its liquidity.
- When conducting trading tests, one might encounter situations where z-tests are more applicable due to the larger sample size, assuming the population standard deviation is known.
- A successful initial coin offering (ICO) might attract attention from investors who perform average daily return tests, using z-tests when the sample size exceeds 30 data points.
- As trade volumes increase, the average return of a stock can be analyzed with z-tests, given that the central finance assumption of normal distribution is met.
- When assessing the performance of a trading strategy, it's important to take into account the average likelihood of results deviating from the expected mean, which can be determined using z-tests.
