To find the critical values for a hypothesis test under the normal distribution for the given significance level (α = 0.00042) and the alternative hypothesis \( H_a: p \neq 0.7 \) (which indicates a two-tailed test), follow these steps:
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Determine the critical values for a two-tailed test: Since this is a two-tailed test, we split the significance level into two tails: \[ \alpha/2 = 0.00042 / 2 = 0.00021 \]
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Find the z-scores corresponding to the critical values: Look for the z-scores that correspond to the cumulative probabilities of \( 0.00021 \) (for the lower tail) and \( 1 - 0.00021 = 0.99979 \) (for the upper tail).
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For \( P(Z < z_1) = 0.00021 \), you can find \( z_1 \) using a standard normal distribution table or a calculator:
- This z-score is approximately \( z_1 \approx -3.75 \) (using z-table or calculator).
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For \( P(Z < z_2) = 0.99979 \):
- This z-score is approximately \( z_2 \approx +3.75 \).
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Critical values: Therefore, the critical values for the test based on the significance level of 0.00042 are:
- \( z \approx -3.75 \) (left tail)
- \( z \approx +3.75 \) (right tail)
In summary, the critical values are approximately: \[ z \approx -3.75 \quad \text{and} \quad z \approx 3.75 \]
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