What Two Assumptions Must Be Met When Using The Z Test

Question

What Two Assumptions Must Be Met When You Are Using The Z Test To Test When conducting a Z test to compare two means, there are two primary assumptions that must be satisfied to ensure the validity of the test results. First, the data should be drawn from populations that are normally distributed or that the sample sizes are sufficiently large, typically over 30, to invoke the Central Limit Theorem. This normality assumption ensures that the sampling distribution of the sample mean differences approximates a normal distribution, which is essential for applying the Z test. Second, the observations must be independent within and between groups; that is, the sampled individuals should not influence each other, and the groups should be independently selected. This independence assumption prevents bias and ensures the applicability of the statistical inference. It is also important to note that when performing a Z test for the difference between two population means, the sample standard deviations (s₁ and s₂) cannot be used in place of the population standard deviations (σ₁ and σ₂) unless the population standard deviations are known. Typically, when the population standard deviations are unknown, a t-test is more appropriate. However, if σ₁ and σ₂ are known and the sample sizes are large enough, the Z test is valid. In practice, since population standard deviations are rarely known, the sample standard deviations are often used in conjunction with the Z test only when the sample sizes are large enough to ensure the sample standard deviations closely estimate the population parameters. Testing the Difference in Social Security Benefits Between Retired and Disabled Workers Suppose we want to test whether the average difference in monthly Social Security benefits between retired workers and disabled workers exceeds $30. To formulate this hypothesis test, we define the null hypothesis as H₀: μ₁ - μ₂ ≤ $30, meaning the difference in means is less than or equal to $30. The alt

Dr. Jack HW Helper · Accepted Answer

What Two Assumptions Must Be Met When You Are Using The Z Test To Test When conducting a Z test to compare two means, there are two primary assumptions that must be satisfied to ensure the validity of the test results. First, the data should be drawn from populations that are normally distributed or that the sample sizes are sufficiently large, typically over 30, to invoke the Central Limit Theorem. This normality assumption ensures that the sampling distribution of the sample mean differences approximates a normal distribution, which is essential for applying the Z test. Second, the observations must be independent within and between groups; that is, the sampled individuals should not influence each other, and the groups should be independently selected. This independence assumption prevents bias and ensures the applicability of the statistical inference. It is also important to note that when performing a Z test for the difference between two population means, the sample standard deviations (s₁ and s₂) cannot be used in place of the population standard deviations (σ₁ and σ₂) unless the population standard deviations are known. Typically, when the population standard deviations are unknown, a t-test is more appropriate. However, if σ₁ and σ₂ are known and the sample sizes are large enough, the Z test is valid. In practice, since population standard deviations are rarely known, the sample standard deviations are often used in conjunction with the Z test only when the sample sizes are large enough to ensure the sample standard deviations closely estimate the population parameters. Testing the Difference in Social Security Benefits Between Retired and Disabled Workers Suppose we want to test whether the average difference in monthly Social Security benefits between retired workers and disabled workers exceeds $30. To formulate this hypothesis test, we define the null hypothesis as H₀: μ₁ - μ₂ ≤ $30, meaning the difference in means is less than or equal to $30. The alt

What Two Assumptions Must Be Met When You Are Using The Z Test To Test

Testing the Difference in Social Security Benefits Between Retired and Disabled Workers

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