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The z-test is a statistical method used to determine if there is a significant difference between the means of two groups. It is commonly employed when the sample size is large, typically greater than 30. The z-test uses the standard normal distribution to compare sample data against a null hypothesis. There are different types of z-tests, which are classified based on the nature of the hypothesis being tested:

• One-sample z-test: Used to determine if the sample mean is significantly different from a known population mean.
• Two-sample z-test: Used to compare the means of two independent groups to see if they come from populations with the same mean.
• Paired z-test: Used to compare the means of two related groups.

The core idea of the z-test is to calculate a z-score, which measures the number of standard deviations the observed statistic is from the mean. The z-score is then used to find the p-value, which indicates the probability of observing the data if the null hypothesis is true.

The standard normal distribution is a type of probability distribution that has a mean of 0 and a standard deviation of 1. It is symmetric about the mean, creating a bell-shaped curve. This the simplest form of normal distribution and is used as the basis for z-tests. Key properties include:

• Mean = 0: It is centered around zero.
• Standard deviation = 1: This makes it a "standard" normal distribution, differing from normal distributions with other means or standard deviations.
• Symmetry: The curve is perfectly symmetrical around the mean.
• Total area under the curve equals 1: Represents the entire population of data.

When performing a z-test, standardized scores (z-scores) are derived from data points to determine their position on the standard normal distribution, helping to ascertain the likelihood of observing such results under the null hypothesis.

In a right-tailed test, the hypothesis test is designed to assess whether a sample mean is greater than a known value. It focuses on values in the right tail of the distribution. Here's how it works:

• Null hypothesis ( H_0 a): The sample mean is less than or equal to the population mean.
• Alternative hypothesis ( H_a a): The sample mean is greater than the population mean.

To find the p-value for a right-tailed test, you look at the area to the right of the observed z-score on the standard normal distribution. This p-value indicates the probability of observing a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.

A two-tailed test is used when the hypothesis test checks for deviations on both ends of a distribution. It allows for the investigation of differences in either direction; that is, it checks if the sample mean is either significantly higher or lower than the population mean. How it works:

• Null hypothesis ( H_0 a): The sample mean is equal to the population mean.
• Alternative hypothesis ( H_a a): The sample mean is not equal to the population mean.

In a two-tailed test, the p-value is calculated by looking at the areas beyond the observed z-score in both tails of the standard normal distribution. This involves doubling the tail probability of the observed z-score's absolute value. This gives a comprehensive view of both possible directions of deviation, making it suitable when there is no specific direction of effect hypothesized.

A left-tailed test is used when the objective is to assess whether the sample mean is less than a specified value. This test focuses on the left tail of the distribution to determine the likelihood of observing a result as extreme as, or more extreme than, the sample. Key components:

• Null hypothesis ( H_0 a): The sample mean is greater than or equal to the population mean.
• Alternative hypothesis ( H_a a): The sample mean is less than the population mean.

The p-value for a left-tailed test is found by calculating the area to the left of the observed z-score on the standard normal distribution. A smaller p-value signifies a greater statistical significance against the null hypothesis, indicating stronger evidence that the sample mean is indeed less than the population mean.