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Confidence Interval Calculators

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 Proportions

Sample Proportion (p̄): Sample Size (n): Population Proportion (p₀): Significance Level (α): Test Type: Left-Tailed Test (H₁: p < p₀) Right-Tailed Test (H₁: p > p₀) Two-Tailed Test (H₁: p ≠ p₀)

Results

Tips about Confidence Interval for Mean when σ is Unknown:

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Definations

• Proportion: The ratio of the number of successes to the total number of trials in a given sample. It is usually denoted as p for the population proportion and \(\bar{p}\) for the sample proportion.
• Alternative Hypothesis (\(H_1\)): The statement that contradicts the null hypothesis. It suggests that there is an effect or a difference. For proportions, it might state that the proportion is not equal to (≠), greater than (>), or less than (<) a specified value.

General Steps

1. State the Null and Alternative Hypotheses:
   • \(H_0: P = P_0 : P ≥ P_0 : P ≤ P_0\)
   • \(H_0: P ≠ P_0 : P < P_0 : P > P_0\)
   
   The inequality sign in the Alternative hypothesis indicates the type of tests. i.e., \(H_1\): \(P ≠ P_0\) (two-tailed), \(P > P_0\) (right-tailed), or \(P < P_0\) (left-tailed)


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


3. Calculate the test Statistcs (\(z_{\bar{p}}\))
   • Calculate the sample proportion \(\bar{p}\) \[\bar{p} = \frac {x}{n}\]
   • Compute the standard error for proportion \(δ_{\bar{p}}\)
   \[δ_{\bar{p}} = \sqrt{\frac{p_0 (1 - p_0)}{n}}\]
   • Calculate the z-test statistcs
   \[z_{\bar{p}} = \frac{\bar{p} - P_0}{\delta_{\bar{p}}}\]

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Input

Standard Error (δ̅p)	--
Z-Test Statistic (z̅p)	--




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:
     \(z_{\alpha}\) is a one-tailed z-critical Value.
     \(z_{\alpha/2}\) is a two tailed z-critical valu

     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 \(z_{\bar{p}}\) to the \(z_{\alpha}\).
   • If | \(z_{\bar{p}}\) | > |\(z_{\alpha}\)|, reject the \(H_0\).
   • If | \(z_{\bar{p}}\) | ≤ |\(z_{\alpha}\)|, 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.
   
   
   
   
   
   Example:
   Senarios:
   • Suppose a company claims that their new training program improves employee productivity by at least 10%. We want to test this claim using a significance level of 0.05.
   
   First identifay the claim from the statement:
   • The claim is stated in \(H_0\); because the senario indicates the greaterthan or equal to statement (at least 10%) that alligned with the null hypothesis.
   
   Decision
   • If the decision is reject \(H_0\): Then the conclusion is "Based on the test, there is enough evidences to reject the claim that the training program does improve productivity by at least 10%.
   
   
   • If the decision is fail to reject \(H_0\): Then the conclusion is "Based on the test, there is not enough evidences to reject the claim that the training program does improve productivity by at least 10%.

Common Errors:

1. Ignoring Assumptions:
   • he test for proportions assumes a sufficiently large sample size for the normal approximation to be valid. Check that \(nP_0\) and \(n(1-P_0)\) are both greater than 5.
2. Not Considering the Effect Size:
   • A small p-value does not necessarily mean the effect is practically significant. Consider the effect size and practical implications.
3. Misinterpreting the the result:
   • Reject \(H_0\) when abosulate value test statistcs is greater than the absoulate value critical value.
4. Sample Size Issues:
   • Small sample sizes can lead to inaccurate conclusions. Ensure the sample size is adequate for the test being conducted.

Additional Tips

1. Check Assumptions:
   • Ensure the sample size is large enough for the normal approximation to hold. Use exact tests (like Fisher’s exact test) for small sample sizes if needed.
2. Use Confidence Intervals:
   • In addition to hypothesis tests, calculate confidence intervals for the proportion to provide a range of plausible values for the true proportion.
3. Understand the Context:
   • Interpret results in the context of the problem. A statistically significant result might not be practically significant.

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