Z-Test

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 for an accurate z-test to be performed.

BREAKING DOWN 'Z-Test'

A one-sample location test, two-sample location test, paired difference test and maximum likelihood estimate are examples of tests that can be conducted as z-tests. 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.

Hypothesis Test

The z-test is 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.

One-Sample Z-Test Example

For example, assume an investor wishes to test whether the average daily return of a stock is greater than 1%. A simple random sample of 50 returns is calculated and has an average of 2%. Assume the standard deviation of the returns is 2.50%. 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 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. 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 greater than 1.96 and concludes the average daily return is greater than 1%.

RELATED TERMS
  1. T-Test

    A statistical examination of two population means. A two-sample ...
  2. Alpha Risk

    The risk in a statistical test that a null hypothesis will be ...
  3. Hypothesis Testing

    A process by which an analyst tests a statistical hypothesis. ...
  4. Central Limit Theorem - CLT

    A statistical theory that states that given a sufficiently large ...
  5. Sample

    A subset containing the characteristics of a larger population. ...
  6. Two-Tailed Test

    A statistical test in which the critical area of a distribution ...
Related Articles
  1. Trading

    What's a T-Test?

    T-Test is a term from statistics that allows for the comparison of two data populations and their means. The test is used to see if the two sets of data are significantly different from one another. ...
  2. Trading

    Hypothesis Testing in Finance: Concept & Examples

    When you're indecisive about an investment, the best way to keep a cool head might be test various hypotheses using the most relevant statistics.
  3. Investing

    What is a Null Hypothesis?

    In statistics, a null hypothesis is assumed true until proven otherwise.
  4. Investing

    Explaining Standard Error

    Standard error is a statistical term that measures the accuracy with which a sample represents a population.
  5. Markets

    How Does Sampling Work?

    Sampling is a term used in statistics that describes methods of selecting a pre-defined representative number of data from a larger data population.
  6. Markets

    What is Systematic Sampling?

    Systematic sampling is similar to random sampling, but it uses a pattern for the selection of the sample.
  7. Markets

    What is a Representative Sample?

    In statistics, a representative sample accurately represents the make-up of various subgroups in an entire data pool.
  8. Markets

    Explaining the Central Limit Theorem

    Central limit theorem is a fundamental concept in probability theory.
  9. Investing

    How Statistical Significance is Determined

    If something is statistically significant, it’s unlikely that it happened by chance.
  10. Markets

    Understanding the Simple Random Sample

    A simple random sample is a subset of a statistical population in which each member of the subset has an equal probability of being chosen.
RELATED FAQS
  1. What assumptions are made when conducting a t-test?

    Learn what a t-test is, and discover the five standard assumptions that are made regarding the validity of sampling and data ... Read Answer >>
  2. What is the relationship between confidence inferrals and a null hypothesis?

    Learn about the relationship between confidence intervals and the null hypothesis in scientific research and empirical experimentation. Read Answer >>
  3. How can a representative sample lead to sampling bias?

    Learn how using representative samples alone is not enough to make sampling bias negligible and why elements such as randomization ... Read Answer >>
  4. What does a strong null hypothesis mean?

    Find out what null hypothesis is and why it is important to the scientific method. See how statisticians and economists use ... Read Answer >>
  5. What's the difference between a representative sample and a convenience sample?

    Learn the difference between convenience sampling and representative sampling and the advantages and disadvantages of each ... Read Answer >>
  6. What is the difference between systematic sampling and cluster sampling?

    Learn about the differences between systematic sampling and cluster sampling, including how the samples are created for each ... Read Answer >>
Hot Definitions
  1. Glass-Steagall Act

    An act the U.S. Congress passed in 1933 as the Banking Act, which prohibited commercial banks from participating in the investment ...
  2. Quantitative Trading

    Trading strategies based on quantitative analysis which rely on mathematical computations and number crunching to identify ...
  3. Bond Ladder

    A portfolio of fixed-income securities in which each security has a significantly different maturity date. The purpose of ...
  4. Duration

    A measure of the sensitivity of the price (the value of principal) of a fixed-income investment to a change in interest rates. ...
  5. Dove

    An economic policy advisor who promotes monetary policies that involve the maintenance of low interest rates, believing that ...
  6. Cyclical Stock

    An equity security whose price is affected by ups and downs in the overall economy. Cyclical stocks typically relate to companies ...
Trading Center