-
Table of Contents
Z-TEST EXAMPLE PROBLEMS WITH SOLUTIONS
When it comes to statistical analysis, the Z-test is a powerful tool used to determine whether there is a significant difference between a sample mean and a population mean. Understanding how to apply the Z-test in real-world scenarios is crucial for making informed decisions based on data. In this article, we will explore some Z-test example problems with solutions to help you grasp the concept better.
Problem 1: Z-Test for Population Mean
Suppose a shoe company claims that the average shoe size of their customers is 9. To test this claim, a random sample of 50 customers is selected, and their shoe sizes are measured. The sample mean is found to be 8.5 with a standard deviation of 1.2.
. Conduct a Z-test at a significance level of 0.05 to determine if the company’s claim is valid.
Solution:
- Null Hypothesis (H0): The average shoe size of customers is 9.
- Alternative Hypothesis (H1): The average shoe size of customers is not 9.
- Calculate the Z-score using the formula: Z = (sample mean – population mean) / (standard deviation / sqrt(sample size)).
- Compare the calculated Z-score with the critical Z-value at the 0.05 significance level.
- If the calculated Z-score falls within the critical region, reject the null hypothesis.
By following these steps and performing the calculations, you can determine whether the company’s claim about the average shoe size of their customers is statistically significant.
Problem 2: Z-Test for Proportion
Imagine a political poll conducted to determine the proportion of voters in favor of a particular candidate. Out of 500 respondents, 300 express support for the candidate. Conduct a Z-test at a significance level of 0.01 to determine if the candidate has majority support among voters.
Solution:
- Null Hypothesis (H0): The proportion of voters in favor of the candidate is 0.5.
- Alternative Hypothesis (H1): The proportion of voters in favor of the candidate is greater than 0.5.
- Calculate the Z-score using the formula: Z = (sample proportion – population proportion) / sqrt((population proportion * (1 – population proportion)) / sample size).
- Compare the calculated Z-score with the critical Z-value at the 0.01 significance level.
- If the calculated Z-score falls within the critical region, reject the null hypothesis.
By applying the Z-test for proportion in this scenario, you can determine whether the candidate enjoys majority support among voters based on the poll results.
Conclusion
The Z-test is a valuable statistical tool that allows researchers and analysts to make informed decisions based on data. By understanding how to apply the Z-test in various scenarios, you can assess the significance of sample statistics and draw meaningful conclusions. Remember to follow the steps outlined in the examples provided to conduct Z-tests effectively and interpret the results accurately.
For further reading on Z-tests and statistical analysis, you can refer to resources such as Statisticshowto.com for comprehensive guides and tutorials.
