When testing hypotheses using proportions, the following requirements must be satisfied to ensure valid results:

1. Random Sampling: The sample must be randomly selected from the population to reduce bias and ensure the results are representative.

2. Independence: Each observation in the sample should be independent. This is typically satisfied if the sample size is less than 10% of the population when sampling without replacement.

3. Sample Size Condition: The sample size must be large enough for the sampling distribution of the proportion to be approximately normal. This is checked using the conditions:

   • $n \cdot p_0 \geq 10$ (expected successes)
   • $n \cdot (1 - p_0) \geq 10$ (expected failures) where $n$ is the sample size and $p_0$ is the hypothesized population proportion.

4. Binary Outcome: The variable being studied should have a binary outcome (e.g., success/failure, yes/no).

5. Known Hypothesized Proportion: The null hypothesis should provide a specific proportion ($p_0$) for comparison.

If these conditions are met, you can use the z-test for proportions to test hypotheses.

Would you like further explanation or an example to illustrate this process?

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Relative Questions:

1. What is the formula for the test statistic in a hypothesis test for proportions?
2. How do you calculate the p-value in a hypothesis test for proportions?
3. Can you explain the difference between one-tailed and two-tailed tests in this context?
4. What are some common errors to avoid when interpreting hypothesis test results for proportions?
5. How does the confidence level affect the interpretation of hypothesis test results?

Tip: Always check the sample size condition for normality before proceeding with a hypothesis test involving proportions.