To determine the critical value related to a hypothesis test, we must first understand the components involved in your question:
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Significance Level (α): You have provided a significance level of 0.042%, which is \( \alpha = 0.00042\) in decimal form.
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Alternative Hypothesis: The alternative hypothesis you specified is \( H_a: p < 0.7 \). This indicates that you are performing a one-tailed test, specifically looking for evidence that the population proportion \( p \) is less than 0.7.
Since you indicated a one-tailed test with a left-sided alternative hypothesis, we will find the critical value corresponding to your specified significance level.
Finding the Critical Value
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Determine the Critical Value: The critical value can be found using the standard normal distribution (Z-distribution) because hypothesis tests for proportions often assume normality for large enough sample sizes.
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Use Z-tables or statistical software: For a significance level of \( \alpha = 0.00042 \), you need to find the z-value such that the area to the left of it is equal to \( 0.00042 \).
To find the critical value:
- Use a z-table, statistical calculator, or statistical software to look for the z-value that corresponds to cumulative probability 0.00042.
Calculation
Using a standard normal distribution:
- You are looking for \( z \) such that \( P(Z < z) = 0.00042 \).
Using standard software or calculators, you could find that:
\[ z \approx -3.12 \]
Thus, the critical value \( z \) for your test is approximately -3.12.
Conclusion
If your calculated test statistic is less than -3.12, you would reject the null hypothesis in favor of the alternative hypothesis \( H_a: p < 0.7 \). Otherwise, you would not reject the null hypothesis. Always ensure to check your data against the assumptions of the test and verify calculations with statistical tools if needed.