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Hypothesis Testing for mean when σ is known

Sample Mean (x̄): Population Mean (μ₀): Sample Size (n): Significance Level (α): Population Standard Deviation (σ): Does the population follow a normal distribution? Yes No Test Type: Left-Tailed Test (H₁: μ < μ₀) Right-Tailed Test (H₁: μ > μ₀) Two-Tailed Test (H₁: μ ≠ μ₀)

Results

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Tips about Hypothesis testing for Mean when σ known:

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Definations

• Hypothesis Testing: A statistical method used to make inferences or draw conclusions about a population based on sample data. It involves testing an assumption (hypothesis) about a population parameter.
• Null Hypothesis (H₀): The default assumption that there is no effect or no difference. It is the hypothesis that the test seeks to nullify.
• Alternative Hypothesis (H₁ or Ha): is a statement that indicates the presence of an effect or difference. It is what you want to test for. For example, H₁ might state that there is a difference between two population means.
• Significance Level (α): is the probability of rejecting the null hypothesis when it is actually true (Type I error). Common significance levels are 0.05, 0.01, and 0.10.
• P-Value: is the probability of observing the test results under the null hypothesis. It helps determine whether to reject the null hypothesis. A p-value less than the significance level indicates strong evidence against H₀.
• Critical Value:is a threshold that defines the boundary for rejecting the null hypothesis. It is based on the significance level and the distribution of the test statistic.

General Steps

1. Formulate Hypotheses:
   • The null and alternative hypothesis for each tail stated as follow as:
   \(H_0:μ = μ₀ : μ ≥ μ₀ : μ ≤ μ₀\)
   \(H_1: μ ≠ μ₀ : μ < μ₀ : μ > μ₀\)
   
   The inequality sign in the Alternative hypothesis indicates the type of tests. i.e., \(H_1\): \(μ₁ ≠ μ₂ \) (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%. This determines the threshold for rejecting H₀.


3. Choose the Test statistic: ,
   • 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 z-test statistsics
   \[ z = \frac{\bar{x} - \mu_0}{\frac{δ}{\sqrt{n}}} \]

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Sample Data

Standard Error (\(SE\))	--
Z-Statistic (\(z\))	--




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:
   • Reject H₀ if the test statistic falls in the rejection region or if the p-value is less than α.
   • Fail to reject H₀ if the test statistic does not fall in the rejection region or if the p-value is greater than α.
   • In general the decision will be made by comparing the \(z_{\bar{x}}\) to the \(z_{\alpha}\).
   If | \(z_{\bar{x}}\) | > |\(z_{\alpha}\)|, reject the \(H_0\).
   If | \(z_{\bar{x}}\) | ≤ |\(z_{\alpha}\)|, 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. Type I Error (False Positive):
   • Rejecting H₀ when it is true. The probability of making this error is α.
2. Type II Error (False Negative):
   • Failing to reject H₀ when H₁ is true. The probability of making this error is β.
3. Misinterpreting the P-Value:
   • The p-value does not measure the probability that H₀ is true, but rather the probability of observing the data if H₀ is true.
4. Overlooking Assumptions:
   • Many tests have assumptions (e.g., normality, equal variances). Violating these assumptions can affect the validity of the test results.
5. Sample Size Issues:
   • Small sample sizes can lead to inaccurate estimates and increased risk of Type II errors.

Additional Tips

1. Understand Your Data:
   • Make sure to understand the characteristics of your data and choose the appropriate test based on the data type and distribution.
2. Check Assumptions:
   • Verify that the data meets the assumptions required for the statistical test you are using.
3. Report Effect Size:
   • Alongside p-values, report the effect size to provide a measure of the magnitude of the observed effect.
4. Be Cautious with the Significance Level:
   • While α = 0.05 is common, adjust the significance level based on the context and consequences of Type I and Type II errors.