Z- Test and P-Value are two statistical tests, but these are two separate things. Where the former is a statistical test that throws light upon whether one should reject the null hypothesis or not, whereas the latter is a probability test signifying there is a probability that the null hypothesis will be rejected.

## Key Takeaways

- Statistical concepts: Z-test is a hypothesis test using the standard normal distribution. At the same time, the p-value represents the probability of observing a test statistic as extreme as the one obtained, assuming the null hypothesis is true.
- Purpose: Z-test is used to compare a sample statistic to a population parameter, while the p-value helps determine the test result’s significance.
- Decision-making: Z-test results in a test statistic (z-score), compared to a critical value; if the z-score is more extreme than the critical value, the null hypothesis is rejected. P-value aids this decision-making process by providing a probability measure.

**Z-Test vs P-Value**

The z-test is a hypothesis testing procedure used when the sample size is large, and the population standard deviation is known. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, and is used for both large and small sample sizes.

A Z-test in statistics is a tool used to determine whether two population means vary even when the variables are known.

A null hypothesis is a general statement stating no relationship between the two measured groups.

## Comparison Table

Parameters of Comparison | P-Value | Z-Test |
---|---|---|

Meaning | The P-Value is the probability of the observations remaining the same or extreme if the null hypothesis is true. | Z-Test describes the deviation from the mean in units of standard deviation. |

Assumptions | The P-Value is the test carried forward assuming the null hypothesis is true. | In the case of the Z-Test, it does not make such assumptions. |

Objective | The objective of this test is to find out whether the null hypothesis should be accepted or not. | The objective of this test is to check if the observations remain the same or not and if the null hypothesis is true. |

Indication of the test | The P-value indicates how unlikely the statistic is. | Whereas the Z-Test indicates how far the mean is. |

## What is Z-Test?

A Z-test in statistics is a tool that is used to determine whether two population means vary even when the variables are known. Moreover, the sample size is large.

The Z-scores are the standard deviation measures; for example, +1.95 or -1.95 denotes how much the test statistic result has deviated from the mean.

There are a few assumptions that are made in the One-Sample Z-test:

- The data are continuous and not discrete.
- The data follow the normal probability distribution.

## What is P-Value?

The P-Value is the probability of the test statistic result being rejected or accepted with an assumption of the null hypothesis being correct.

To find out the p-value in one’s statistic:

- Look up the statistic on the appropriate distribution.
- Find the probability that the mean is beyond your test statistic.
- If the hypothesis is less than the alternative, find the probability of the mean being less than your test statistic. This is the p-value.

**Main Differences Between Z-Test and P-Value**

**Meaning**

The P-Value is the probability of obtaining a test statistic result equal to or as extreme as a result observed in the experiment, assuming that the null hypothesis is true.

Whereas the Z-Test is the test used to determine whether the mean of a population is more significant than, less than, or equal to a specific value.

**Null Hypothesis**

In the case of the P-Value, the null hypothesis is assumed to be accurate, based on which the test statistic result observed in the experiment is checked to see if the result is the same or extreme as it was kept before.

**Alternative Hypothesis**

In the P-Value, the alternative hypothesis is the crucial statement that the experimenter wants to conclude in the experimental test if the data allows it.

**Limitations**

Moreover, the p-value tends to be concluded as significant or non-significant based on the p-value being less than or equal to 0.5, which is not the case with Z-Test. However, there are a few limitations to using the Z-Test.

The sample size may range from a small number to several hundred; if the data is discrete with at least five unique values, one may ignore the continuous variable assumption.

**Results**

Suppose the p-value is very small as compared to the threshold value that was previously chosen, known as the significant level (commonly 5% or 1%). In that case, it suggests that the observed data is inconsistent with the assumption that the null hypothesis is true. Thus, the hypothesis must be rejected, and the alternative hypothesis must be accepted.

For example:

- p < 0.1, the hypothesis is rejected
- 0.1<p<0.5, the hypothesis may or may not be rejected
- p>0.1, the hypothesis is accepted

In Z-Test, for example, The critical Z-Score values when using a 95% confidence level, -1.96 and +1.96 standard deviations.

If the Z score falls outside that range (for example, -2.5 or +5.4), the pattern exhibited is probably too unusual to be just another version of random chance, and the p-value will be small to reflect this.

**References**