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HT for µ (σ Known) HT for µ (σ Unknown) HT for µ1 - µ2 (σ Known)

HT for µ1 - µ2 (σ Unknown) HT for P HT for (P1-P2)

Hypothesis Testing for the difference between mean when σ's are Unknown

Sample Mean 1 (x̄₁): Sample Mean 2 (x̄₂): Sample Size 1 (n₁): Sample Size 2 (n₂): Significance Level (α): Sample Standard Deviation 1 (s₁): Sample Standard Deviation 2 (s₂): 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) Variance Assumption: Equal Variances Unequal Variances Does the population follow a normal distribution? Yes No

Results

Tips about Hypothesis Testing for Mean when σ is Unknown:

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Definations

• Hypothesis Testing: A statistical method to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis.
• Null Hypothesis (H₀): The hypothesis that there is no effect or no difference. For two means, it often states that the means are equal (μ₁ = μ₂), less than or equal to ((μ₁ ≤ μ₂), or greater than or equla to (μ₁ ≥ μ₂) .
• Alternative Hypothesis (H₁): The hypothesis that there is an effect or a difference. For two means, it can state that the means are not equal (μ₁ ≠ μ₂), less than (μ₁ < μ₂), or greater than (μ₁ > μ₂).
• Test Statistic: A standardized value used to determine the probability of observing the test results under the null hypothesis. For unknown population standard deviations, this is usually the t-statistic.
• Degrees of Freedom (df): A parameter that influences the shape of the t-distribution.
• Significance Level (α): The probability of rejecting the null hypothesis when it is actually true. Common values are 0.05, 0.01, and 0.10.

General Steps

1. Formulate Hypotheses:
   • \(H_0: μ₁ = μ₂ : μ₁ ≥ μ₂ : μ₁ ≤ μ₂\)
   • \(H_1: μ₁ ≠ μ₂ : μ₁ < μ₂ : μ₁ > μ₂\)
   
   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)


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


3. Calculate the test Statistcs (\(t_{\bar{x_1}-{x_2}}\)),
   • For equal variances: Use pooled variance.
   \[ p_o = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \] Then the standard error will be \[ δ_{\bar{x}_1 - \bar{x}_2} = p_o \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} \] and the degree of fredom is as follows: \[ df = n_1 + n_2 - 2 \]
   • For unequal variances: Use the separate variances of each sample.
   \[ δ_{\bar{x}_1 - \bar{x}_2} = \sqrt{\left( \frac{s_1^2}{n_1} \right) + \left( \frac{s_2^2}{n_2} \right)} \] and the degree of freedom will be used the Satterthwaite approximation. \[ df \approx \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \]
   • Calculate the t-test statistcs \(t_{\bar{x_1}-{x_2}}\),
   \[t_{\bar{x_1}-{x_2}} = \frac {({{\bar{x_1}-{\bar{x_2}}}}) - ({μ_1}-{μ_2})}{\delta_{\bar{x_1}-{x_2}}}\]

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Sample 1	Sample 2
Equal Variance Not Equal Variance

Standard Error (δ̅x1 - ̅x2)	--
Degrees of Freedom	--
t-Statistic (\(t_{\bar{x_1}-{x_2}}\))	--




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

   • In the table below, I’ve listed common significance levels (10%, 5%, 1%) along with their corresponding degrees of freedom (df) and \( t_{\alpha/2} \) and \( t_{\alpha} \) values:

     α	df	\( t_{\alpha/2} \)	\( t_{\alpha} \)
     0.1	30	1.697	1.310
     0.05	30	2.042	1.697
     0.01	45	2.750	2.015
     		N/A	N/A

     where:
     α is a significance level.
     df is the degree of freedom.
     \(t_{\alpha}\) is a one-tailed t-critical Value.
     \(t_{\alpha/2}\) is a two tailed t-critical value

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



5. Decision:
   The decision will be made by comparing the \(t_{\bar{x_1}-\bar{x_2}}\) to the \(t_{\alpha}\).
   • If | \(t_{\bar{x_1}-\bar{x_2}}\) | > |\(t_{\alpha}\)|, reject the \(H_0\).
   • If | \(t_{\bar{x_1}-\bar{x_2}}\) | ≤ |\(t_{\alpha}\)|, 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. Incorrect Hypothesis Formulation:
   • Misstating the null and alternative hypotheses can lead to incorrect conclusions.
2. Misinterpreting the Significance Level:
   • Confusing the significance level (α) with the p-value.
3. Using the Wrong Degrees of Freedom:
4. Misinterpretation of Results:
   • Especially critical in the case of unequal variances.
5. Ignoring Assumptions:
   • Not checking if the assumptions (normality, independence, etc.) are met can invalidate the test results.

Additional Tips

1. Normality Assumption: For small samples (n < 30):
   • The population should be normally distributed. For larger samples, the Central Limit Theory generally allows for normality.