Z-TEST GROUP 1 AÑONUEGO, CYRUS OBISPO AREGLADO, OSABEL JAMES HANTIC, JAN MARU CARANDANG, JERICH VILLAR, ANTHONY OBJECTIVES • MEANING • TECHNICAL TERMS OF Z-TEST • PURPOSE • WHEN SHOULD YOU USE A Z-TEST? WHAT IS Z-TEST? • When the variances are known and the sample size is large, a z-test is a statistical test that is used to assess whether two population means differ from one another. • In order to execute a precise z-test, the test statistic is expected to have a normal distribution, and nuisance variables like standard deviation should be known. KEY TAKEAWAYS • A z-test is a statistical test to determine whether two population means are different when the variances are known and the sample size is large. • A z-test is a hypothesis test in which the z-statistic follows a normal distribution. • A z-statistic, or z-score, is a number representing the result from the z-test. • Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size. • Z-tests assume the standard deviation is known, while t-tests assume it is unknown. TECHNICAL TERMS 1. Null Hypothesis (H.): The null hypothesis in a Z-test represents the assumption of no significant difference between the sample statistic (e.g., sample mean or proportion) and the population parameter. It is the hypothesis that is tested and either accepted or rejected based on the results of the Z-test. 2. Alternative Hypothesis (H.): The alternative hypothesis in a Z-test specifies the alternative claim or hypothesis, which suggests that there is a significant difference between the sample statistic and the population parameter. It is considered when the evidence from the Z-test suggests rejecting the null hypothesis. TECHNICAL TERMS 3. Significance Level (a): The significance level, denoted as (a), is the predetermined probability threshold used to determine the critical region or critical values in hypothesis testing. It represents the maximum acceptable probability of making a Type I error (rejecting the null hypothesis when it is true). Commonly used values for (a) are 0.05 (5%) or 0.01 (1%). 4. Z-score: The Z-score is a standardized score that measures how many standard deviations a sample statistic is away from the population parameter. In a Z-test, it is calculated by subtracting the population parameter from the sample statistic and dividing it by the standard deviation of the population (or known standard deviation). The Z-score is then compared to critical values to determine the statistical significance of the result. TECHNICAL TERMS 5. Critical Region: The critical region is the range of values in the tail(s) of the standard normal distribution that corresponds to the rejection of the null hypothesis. If the Z-score falls within the critical region, it indicates that the sample statistic is significantly different from the population parameter, leading to the rejection of the null hypothesis. 6. Critical Values: Critical values are specific values on the standard normal distribution that define the boundaries of the critical region. They correspond to the chosen significance level (a) and are used to compare the calculated Zscore in the hypothesis test. If the calculated Z-score exceeds the critical value, the null hypothesis is rejected Do you know the population standard deviation, σ? NO YES Is the sample size above 30? YES NO Use the Z-test Use the T-Test Use the T-Test. PURPOSE OF Z-TEST The purpose of using a Z-test in statistics is to make inferences about a population based on a sample when certain conditions are met. Here are the main purposes of using a Z-test: 1. Hypothesis Testing: The Z-test allows for the testing of hypotheses about population parameters, such as population means or proportions. It helps determine whether there is a significant difference between the sample statistic and the hypothesized population parameter. 2. Comparing Sample Mean to Population Mean: The Z-test is commonly used to compare the mean of a sample to a known population mean when the population standard deviation is known. It helps assess whether the observed difference between the sample mean and the population mean is statistically significant. 3. Comparing Sample Proportion to Population Proportion: The Z-test is also used to compare the proportion of successes or events in a sample to a known population proportion. It helps determine if the difference between the observed proportion and the hypothesized population proportion is statistically significant. 4. Confidence Interval Estimation: The Z-test can be employed to construct confidence intervals, which provide a range of plausible values for the population parameter. By calculating the Z-score and using it to determine the margin of error, a confidence interval can be created to estimate the true population parameter. 5. Quality Control: Z-tests are utilized in quality control processes to assess whether a production or manufacturing process is operating within specified tolerances. By comparing sample statistics to target values or acceptable ranges, Z-tests help identify deviations from the desired quality. WHEN SHOULD YOU USE A Z-TEST? The Z-test is commonly used in statistics under the following circumstances: 1. Large Sample Size: The Z-test is suitable when the sample size is large (typically greater than 30). With a large sample size, the sampling distribution of the sample mean tends to approximate a normal distribution, which allows for the application of the Z-test. 2 Known Population Standard Deviation: The Z-test assumes that the population standard deviation is known. This assumption is often met in situations where historical data or prior knowledge of the population provides a reliable estimate of the standard deviation 3. Testing Hypotheses: The Z-test is used for hypothesis testing to evaluate whether a sample mean or proportion is significantly different from a known population mean or proportion. It helps determine if the observed difference is statistically significant or occurred by chance. 4. Normality Assumption: The Z-test assumes that the underlying population follows a normal distribution. If the population is not normally distributed, the Z-test may still be used when the sample size is large enough due to the Central Limit Theorem, which states that the sampling distribution of the sample mean becomes approximately normal with a large sample size. 5. Comparing Means or Proportions: The Z-test is applied when comparing two means or proportions from independent samples, assuming the sample sizes are large and the population standard deviations are known. Problem A: A manufacturers claims that their circuit breaker have a mean short circuit current of 5000 A. To test this claim, an electrical engineer takes a sample of 50 circuit breakers and measures the short circuit current they can withstand. The sample mean short circuit current is 5100 A, with a standard deviation of 600 A. Test the company’s claim at a 5% significance level. Problem B: An electrical engineer wants to determine if the power transmission cable meets the specified insulation resistance requirements. The engineer takes a sample of 40 cables and measures their insulation resistance. The sample mean insulation resistance is 500 MΩ, with a standard deviation of 50 MΩ. The engineer wants to test if the mean insulation resistance is significantly greater than 480 MΩ. Test the hypothesis at a 5% significance level.
