What is a critical value of Z?

As a domain expert in statistics, I would like to clarify the concept of a critical value of Z, which is an important concept in hypothesis testing and statistical inference. A critical value is a threshold that helps us determine whether to reject or fail to reject a null hypothesis based on the test statistic. In the context of Z-tests, which are a type of statistical test used when the sampling distribution of the test statistic can be approximated by a normal distribution, the critical value of Z plays a pivotal role. ### Understanding the Z-Score and Critical Values The Z-score is a standard score that indicates how many standard deviations an element is from the mean. It is calculated using the formula: \[ Z = \frac{(X - \mu)}{\sigma} \] Where: - \( X \) is the value of the element, - \( \mu \) is the population mean, - \( \sigma \) is the standard deviation of the population. ### When to Use Critical Values of Z Critical values of Z are used in hypothesis testing when the following conditions are met: 1. Population Parameters: The population standard deviation is known or can be assumed with a high degree of certainty. 2. Sampling Distribution: The sampling distribution of the statistic (e.g., the mean) is normally distributed or can be approximated by a normal distribution, particularly when sample sizes are large due to the Central Limit Theorem. 3. Hypothesis Testing: The test is a two-tailed or one-tailed test concerning the mean of a population. ### Calculating and Interpreting Critical Values The critical value of Z, sometimes denoted as \( z_\alpha \), is determined by the level of significance \( \alpha \), which is the probability of making a Type I error (rejecting a true null hypothesis). The value of \( z_\alpha \) is found such that the area in the tail(s) of the standard normal distribution equals \( \alpha \). For example, if \( \alpha = 0.10 \), then \( z_{0.10} = 1.28 \) indicates that 10% of the distribution's area lies beyond 1.28 standard deviations from the mean. ### Steps in Hypothesis Testing Using Z-Score 1. **State the Null and Alternative Hypothesis**: Clearly define \( H_0 \) and \( H_1 \) based on the research question. 2. Choose a Significance Level: Select \( \alpha \), which is the threshold for deciding statistical significance. 3. Calculate the Test Statistic: Compute the Z-score for the sample data. 4. Determine the Critical Value: Find the \( z_\alpha \) that corresponds to the chosen \( \alpha \) level. 5. Make a Decision: Compare the calculated Z-score to the critical value(s) to decide whether to reject \( H_0 \) or not. ### Example Suppose we are testing if the average weight of a certain brand of apples is different from 150 grams. We set \( H_0: \mu = 150 \) and \( H_1: \mu \neq 150 \) with \( \alpha = 0.05 \) for a two-tailed test. If our sample mean is 155 grams with a standard deviation of 10 grams and a sample size of 36, the Z-score would be: \[ Z = \frac{(155 - 150)}{\frac{10}{\sqrt{36}}} = 5 \] Using a standard normal distribution table or calculator, we find that the critical value for \( \alpha = 0.05 \) in a two-tailed test is approximately \( z_{0.025} = 1.96 \). Since our Z-score (5) is greater than the critical value (1.96), we reject the null hypothesis. ### Conclusion Understanding when and how to use critical values of Z is crucial for conducting valid statistical tests. It is important to ensure that the assumptions of the Z-test are met and to interpret the results within the context of the research question. The critical value of Z acts as a benchmark that helps researchers make informed decisions about the validity of their hypotheses.

A critical value of z is sometimes written as za, where the alpha level, a, is the area in the tail. For example, z.10=1.28. When are Critical values of z used? A critical value of z (Z-score) is used when the sampling distribution is normal, or close to normal.Jan 15, 2018