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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 Two-Sample Proportions

Sample Proportion 1 (p̄₁): Sample Proportion 2 (p̄₂): Sample Size 1 (n₁): Sample Size 2 (n₂): Hypothesized Difference (p₁ - p₂): Significance Level (α): Test Type (Choose): Left-Tailed Test (H₁: p₁ < p₂) Right-Tailed Test (H₁: p₁ > p₂) Two-Tailed Test (H₁: p₁ ≠ p₂)

Results

Please enter all input values!

Tips about Confidence Interval for Mean when σ is Unknown:

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Definations

• Hypothesis testing for the difference between proportions involves comparing the proportions of two different groups to determine if there is a statistically significant difference between them. This is commonly used in experiments and surveys to compare success rates, preferences, and other binary outcomes between two groups.

General Steps

1. State the Null and Alternative Hypotheses:
   • \(H_0: p_1 = p_2 : p_1 ≥ p_2 : p_1 ≤ p_2\)
   • \(H_0: p_1 ≠ p_2 : p_1 < p_2 : p_1 > p_2\)
   
   The inequality sign in the Alternative hypothesis indicates the type of tests. i.e., \(H_1\): \(p_1 ≠ p_2\) (two-tailed), \(p_1 > p_2\) (right-tailed), or \(p_1 < p_2\) (left-tailed)


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


3. Calculate the test Statistcs (\(z_{\bar{p_1}-{p_2}}\)),
   • Calculate the sample proportions \(\bar{p_1}\) and \(\bar{p_2}\) \[\bar{p}_1 = \frac {x_1}{n_1}\] \[\bar{p_2} = \frac {x_2}{n_2}\]
   • Compute the pooled proportion \(P_o\)
   We use the pooled proportion \(P_o\) for computing the standard error because the actual values of p1 and p2 are unknown. The pooled proportion provides a single best estimate of the common population proportion under the null hypothesis, facilitating a more accurate calculation of the standard error.
   
   \(P_o\) is found using the following equation: \[P_o = \frac {{x_1}+{x_2}}{{n_1}+{n_2}}\]
   • Compute the standard error for proportion \(δ_{\bar{p_1}-{p_2}}\)
   \[δ_{\bar{p_1}-{p_2}} = \sqrt{(P_o)(1-P_o)(\frac{1}{n_1}+\frac{1}{n_2})}\]
   • Calculate the z-test statistcs \(z_{\bar{p_1}-{p_2}}\)
   \[z_{\bar{p_1}-{p_2}} = \frac {({{\bar{p_1}-{\bar{p_2}}}}) - ({p_1}-{p_2})}{\delta_{\bar{p_1}-{p_2}}}\]

   where:
   \(x_1\) is a number of success in sample 1.
   \(x_2\) is the number of succes in sample 2.
   \(n_1\) is sample size 1.
   \(n_2\) is sample size 2.
   \(P_o\) is a pooled proportion.


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Input

Pooled Proportion (p̂)	--
Standard Error (\(δ_{\bar{p_1}-{p_2}}\))	--
Z-Test Statistic (\(z_{\bar{p_1}-{p_2}}\))	--




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:
     \(z_{\alpha}\) 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.

Common Errors:

1. Incorrect Hypothesis Formulation:
   • Ensure the null and alternative hypotheses are correctly stated.
2. Sample Size Issues:
   • Small sample sizes can lead to inaccurate results. Ensure adequate sample sizes to detect a difference.
3. Assumption Violations:
   • The test assumes the samples are independent and randomly selected. Ensure these conditions are met.
4. Misinterpretation of Results:
   • Understand the difference between statistical significance and practical significance. A statistically significant result may not always be practically important.
5. Ignoring Confidence Intervals:
   • Confidence intervals provide additional information about the estimate's precision. Always consider them in conjunction with hypothesis tests.

Additional Tips

1. Use a Two-Tailed Test by Default:
   • Unless you have a specific reason to use a one-tailed test, two-tailed tests are generally safer and more conservative.
2. Check Assumptions:
   • Verify that the assumptions for hypothesis testing (independence, random sampling, and sufficient sample size) are satisfied before proceeding.
3. Report Effect Sizes:
   • Along with p-values, report the effect size to convey the magnitude of the difference between proportions.
4. Consider Context:
   • Interpret the results within the context of the study. Consider the practical implications and limitations.