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Hypothesis Testing for the Difference Between Means (σ Known)

Sample Mean 1 (x̄₁): Sample Mean 2 (x̄₂): Sample Size 1 (n₁): Sample Size 2 (n₂): Population Standard Deviation 1 (σ₁): Population Standard Deviation 2 (σ₂): Significance Level (α): Hypothesis Difference (μ₁ - μ₂): 0 Not equal to 0 Specify the difference (μ₁ - μ₂): Test Type: Left-Tailed Test (H₁: μ₁ < μ₂) Right-Tailed Test (H₁: μ₁ > μ₂) Two-Tailed Test (H₁: μ₁ ≠ μ₂) Does the population follow a normal distribution? Yes No

Results

Tips about Hypothesis testing for Mean when σ known:

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Definations

• Hypothesis Testing is a statistical method used to make decisions about the population parameters based on sample data. It involves formulating a hypothesis, collecting data, and using statistical analysis to determine whether to reject the null hypothesis.

General Steps

1. Formulate Hypotheses:
   • Null Hypothesis (H₀): A statement of no effect or no difference. It is the hypothesis that the researcher seeks to test
   • Alternative Hypothesis (H₁ or Ha): A statement that contradicts the null hypothesis. It represents the effect or difference the researcher wants to prove
   • Therefore, the null and alternative hypothesis stated as follow as:
   \(H_0: μ₁ = μ₂ : μ₁ ≥ μ₂ : μ₁ ≤ μ₂\)
   \(H_1: μ₁ ≠ μ₂ : μ₁ < μ₂ : μ₁ > μ₂\)
   
   The inequality sign in the Alternative hypothesis indicates the type of tests. i.e., \(H_1\): \(μ₁ ≠ μ₂ \) (two-tailed), \(μ₁ > μ₂ \) (right-tailed), or \(μ₁ < μ₂ \) (left-tailed).
   
   
   
   • If you want to show the hypothesised difference (μ₁ - μ₂), let say the differance is 0 then you can display as folows:
     \(H_0\): μ₁ - μ₂ = 0, : μ₁ - μ₂ ≥ 0, : μ₁ - μ₂ ≤ 0
     \(H_1\): μ₁ - μ₂ ≠ 0, : μ₁ - μ₂ < 0, : μ₁ - μ₂ > 0


2. Specify the significance Level (α)
   • The common α's are 10%, 5%, and 1%.


3. Calculate the test Statistcs (\(z_{\bar{x_1}-{x_2}}\)),
   • Standard error \( δ_{\bar{x}_1 - \bar{x}_2}\): first compute the standard error for the differance between mean.
   \[ δ_{\bar{x}_1 - \bar{x}_2} = \sqrt{\left( \frac{σ₁^2}{n_1} \right) + \left( \frac{σ₂^2}{n_2} \right)} \]
   • Calculate the test statistcs \(z_{\bar{x_1}-{x_2}}\),
   \[z_{\bar{x_1}-{x_2}} = \frac {({{\bar{x_1}-{\bar{x_2}}}}) - ({μ_1}-{μ_2})}{\delta_{\bar{x_1}-{x_2}}}\] Often, the hypothesize differance between mean (μ₁ - μ₂) are zero.

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Standard Error and Z-Test Statistic Calculator

Sample 1	Sample 2

Standard Error (\( δ_{\bar{x}_1 - \bar{x}_2}\))	--
Z-Statistic (\(z_{\bar{x_1}-{x_2}}\))	--




4. Determine the critical value of z for the Desired Confidence Level:

   • Use the table below to find Z-critical values based on significance levels (α) for both two-tailed and one-tailed tests:

     Significance Level (α)	\(z_{\alpha/2}\)	\(z_{\alpha}\)
     0.1	1.645	1.282
     0.05	1.960	1.645
     0.01	2.576	2.326
     	N/A	N/A

     where:
     α is a ssignificance level.
     \(z_{\alpha}\) is a one-tailed z-critical Value.
     \(z_{\alpha/2}\) is a two tailed z-critical value

     Feel free to provide a signifcance level in a given space, then I’ll calculate the corresponding \(z_{\alpha/2}\) and \(z_{\alpha}\) for you only!



5. Decision:
   The decision will be made by comparing the \(z_{\bar{x_1}-\bar{x_2}}\) to the \(z_{\alpha}\).
   • If | \(z_{\bar{x_1}-\bar{x_2}}\) | > |\(z_{\alpha}\)|, reject the \(H_0\).
   • If | \(z_{\bar{x_1}-\bar{x_2}}\) | ≤ |\(z_{\alpha}\)|, fail to reject the \(H_0\).
6. Draw the conclusin:
   • When interpreting the results of a hypothesis test, it’s essential to consider both the claim being tested and the decision made. Here’s a step-by-step guide to drawing conclusions based on the decision to reject or not reject the null hypothesis (\(H_0\)):
   • Follow this step to draw a conclusion
   Identify the Claims:
   • Null Hypothesis \(H_0\) This represents the default position or the claim being tested.
   • Alternative Hypothesis \(H_1\) This is the claim you are trying to find evidence for.
   
   
   Make a Decisions:
   • Reject \(H_0\): Evidence suggests that \(H_0\) is not true.
   • Do Not Reject \(H_0\): Evidence is insufficient to conclude \(H_0\) is not true.
   
   
   Draw a conclusion:
   When the claim is \(H_0\):
   • Reject \(H_0\): There is evidence to rejcet the claim.
   • Do Not Reject \(H_0\): There is not enough evidences to rejct the claim.
   
   When the claim is is \(H_1\):
   • Reject \(H_0\): There is evidence to support the claim.
   • Do Not Reject \(H_0\): There is not enough evidences to support the claim.

Common Errors:

1. Using the Wrong Test:
   • Different tests are appropriate for different types of data and hypotheses. Using the wrong test can lead to incorrect conclusions.
2. Misinterpreting the Significance Level:
   • Confusing the significance level (α) with the p-value.
3. Ignoring Assumptions:
   • Most statistical tests have underlying assumptions (e.g., normality, equal variances). Ignoring these assumptions can affect the validity of the test results.

Additional Tips

1. Check Assumptions:
   • Verify that the data meets the assumptions of the chosen test (e.g., normal distribution, homogeneity of variances).
2. Effect Size Calculation:
   • Measure the magnitude of the effect or difference, which provides more information than just the statistical significance.