A Z-Test is a statistical test used to determine if there is a significant difference between the mean of a sample and a known population mean. It is used to test the hypothesis that the sample mean is different from the population mean.

Example 1:

A company wants to determine if the average salary of their employees is different from the national average salary for their industry. They take a sample of 50 employees and calculate the mean salary of the sample to be $50,000. The national average salary for the industry is $48,000. The company wants to determine if the difference between the sample mean and the population mean is significant.

To conduct a Z-Test, the company would first need to calculate the standard deviation of the sample. The standard deviation is a measure of how spread out the data is. If the standard deviation is small, it means that the data points are close to the mean, while a large standard deviation indicates that the data points are more spread out.

Next, the company would need to determine the Z-score, which is the number of standard deviations that the sample mean is from the population mean. To calculate the Z-score, the company would subtract the population mean from the sample mean and divide the result by the standard deviation of the sample.

In this example, the Z-score would be calculated as follows:

Z-score = (50,000 – 48,000) / (standard deviation of the sample)

The company would then use a Z-table to determine the probability of getting a result this extreme if the sample mean and the population mean are the same. If the probability is low, it indicates that the difference between the sample mean and the population mean is statistically significant.

Example 2:

A high school teacher wants to determine if the average test scores of her students are significantly different from the average test scores of students in the district. She takes a sample of 20 students from her class and calculates the mean test score to be 75. The district average test score is 80. The teacher wants to determine if the difference between the sample mean and the population mean is significant.

To conduct a Z-Test, the teacher would first need to calculate the standard deviation of the sample. She would then calculate the Z-score as follows:

Z-score = (75 – 80) / (standard deviation of the sample)

The teacher would then use a Z-table to determine the probability of getting a result this extreme if the sample mean and the population mean are the same. If the probability is low, it indicates that the difference between the sample mean and the population mean is statistically significant.

In conclusion, a Z-Test is a statistical test used to determine if there is a significant difference between the mean of a sample and a known population mean. It is used to test the hypothesis that the sample mean is different from the population mean. To conduct a Z-Test, the standard deviation of the sample and the Z-score must be calculated, and the probability of getting a result this extreme if the sample mean and the population mean are the same must be determined using a Z-table.