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## 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.

### Explaining '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.

### Z Test FAQ

#### How do you find the Z test?

z = (x – μ) / σ In a situation where the test score is 190. The mean (μ) of the test is 150 and the standard deviation (σ) is 25. Assuming a normal distribution, the z score is: z = (x – μ) / σ

#### How do you perform a Z test?

How to run a Z Test? Highlight the null and alternate hypothesis. Pick an alpha level. Discover the critical value of z in a z table. Calculate the z test statistic (see below). Compare the test statistic to the critical z value and make a decision to either support or reject the null hypothesis.

#### What is difference between z test and t test?

Z-tests are statistical and can compare population means to a sample's. T-tests are used to prove a hypothesis, but their most important use is to determine if two independent sample groups are statistically different.

#### What are the conditions for a two sample z test?

The conditions for the test procedure, called the two-proportion z-test are:Each population must be sampled using a simple random sampling. Independent samples. Each sample must succeed and fail 10 times respectively.

#### Why do we use t test instead of Z test?

Z-tests are statistical and can compare population means to a sample's. T-tests are used to prove a hypothesis, but their most important use is to determine if two independent sample groups are statistically different.

#### Why are two sample z procedures hardly ever used?

The two sample z-test is rarely used because the two population standard deviations σ 1 and σ 2 are not usually known. In place of that, we use sample standard deviations and the t-distribution.

#### How do you interpret t test results?

The P-value should be compared to the α significance level stated earlier. Reject the null hypothesis if it is less than α. Fail to reject the null hypothesis if otherwise. Rejecting the null hypothesis means your alternative hypothesis is correct, and that the data is significant.

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