What Is a Z-Test?

A Z-test compares one mean against a hypothesized value. It tests whether two means are the same. The data must approximately fit a normal distribution, or the test won't work. Parameters such as variance and standard deviation should 𒉰be calculated when performing a Z-tes🧔t.

Hypothesis Testing

The 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, per the 澳洲幸运5官方开奖结果体彩网:central limit theorem, the samples are considered to be approximately normally distributed with larger samples. For a Z-test to be effective, the population must be normally distributed, and the samples must have the same variance. All data points should be independent of one another.

When conducting a Z-test, the null and alternative hypotheses and alpha level should be stated. The Z-score should be calculated, and the results and conclusion represent how many standard deviations above or below the mean population a score derive🦄d 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-te🦩sts are best performed when an experiment has a small sample size. Also, T-tests assume the standard deviation is u🅰nknown, 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 Example

Assume an inv𒁃estor wishes to test whether the average daily return of a stock is greater than 3%. A random sam💙ple of 50 returns is calculated with an average of 2%. Assume the standard deviation of the returns is 2.5%. Therefore, the null hypothesis is when the average, or mean, equals 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 澳洲幸运5官方开奖结果体彩网:two-tailed test. There are 0.025% of the samples in each tail, and the alpha has a critical value of 1.96 or -1.96. If "Z" is greater than 1.96 or less than -1.96, the null hypothesis is rejecte🐽d.

The value for Z is calculated by subtracting the value of the average daily return selected for the test, or 3% in this case, from the observed average of the samples. Next, divide the resulting value by the 澳洲幸运5官方开奖结果体彩网:standard deviation divid๊ed by the square root of the number of observed values.

Therefore, the test statistic is:

(0.02 - 0.03) ÷ (0.025 ÷ √ 50) = -2.83

The investor rejects the null hypothesis since z is less than -1.96 and concludes that the average daily return is less than 3%.

The Bottom Line

A Z-test is used in hypothesis testing to evaluate whether a finding or association is statistically significant. In particular, it tests whether two means are the same. A Z-test can only be used if the population standard deviation is known and the sample siꦜze is 30 data points or larger. Otherwise, a T-test should be employed.