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Hypothesis Testing for mean when σ is Unknown

Sample Mean (x̄): Population Mean (μ₀): Sample Size (n): Significance Level (α): Sample Standard Deviation (s): Test Type: Left-Tailed Test (H₁: μ < μ₀) Right-Tailed Test (H₁: μ > μ₀) Two-Tailed Test (H₁: μ ≠ μ₀) 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 used to make decisions about the population based on sample data.
• Null Hypothesis (H₀): A statement that there is no effect or no difference, and it is what we seek to test against. It usually contains the equality (e.g., H₀: μ = μ₀), less than or equal to ((μ ≤ μ₀), or greater than or equla to (μ ≥ μ₀) .
• Alternative Hypothesis (H₁): A statement that indicates the presence of an effect or difference. It is what we want to provide evidence for (e.g., H₁: μ ≠ μ₀, μ < μ₀, or μ > μ₀).
• 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.
• Critical Value: The threshold or cut-off value that defines the boundary of the rejection region for the null hypothesis.

General Steps

1. Formulate Hypotheses:
   • \(H_0: μ = μ₀ : μ ≥ μ₀ : μ ≤ μ₀\)
   • \(H_1: μ ≠ μ₀ : μ < μ₀ : μ > μ₀\)
   
   The inequality sign in the Alternative hypothesis indicates the type of tests. i.e., \(H_1\): \(μ ≠ μ₀ \) (two-tailed), \(μ > μ₀ \) (right-tailed), or \(μ < μ₀ \) (left-tailed)


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


3. Calculate the test Statistcs (\(t_{\bar{x}}\)),
   • Depends on the sample size, data distribution, and whether the population standard deviation is known.
   • Common tests include Z-test, t-test, chi-square test, and ANOVA.
   • In this case we will use t-test statistsics
   \[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \] and the degree of fredom is as follows: \[ df = n-1 \]

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Standard Error and T-Test Statistic Calculator (One-Sample Test)

Sample Data

Standard Error (\(SE\))	--
T-Statistic (\(t\))	--
Degrees of Freedom (\(df\))	--




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 ssignificance 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:
   Compare the test statistic \(t_{\bar{x}}\) to the critical value \(t_{\alpha}\) or compare the p-value to the significance level.
   • Reject or fail to reject the null hypothesis based on this comparison.
   • If | \(t_{\bar{x}}\) | > |\(t_{\alpha}\)|, reject the \(H_0\).
   • If | \(t_{\bar{x}}\) | ≤ |\(t_{\alpha}\)|, 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. Misinterpreting p-value:
   • The p-value is not the probability that H₀ is true, but the probability of observing the data given that H₀ is true.
4. Confusing Statistical Significance with Practical Significance:
   • A result can be statistically significant but not practically meaningful.
5. Ignoring Assumptions:
   • Many tests assume normality, equal variances, or independent samples. Violating these assumptions can lead to incorrect conclusions.

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.
2. Understanding the Context:
   • Before performing any test, understand the context and research question to choose the correct test and hypotheses.
3. Report Effect Sizes:
   • Along with p-values, report effect sizes to show the magnitude of the difference or effect.