To find the critical values for a hypothesis test where the significance level (\(\alpha\)) is 0.042% and the alternative hypothesis is two-tailed (\(p \neq 0.7\)), we need to first convert the significance level into a decimal:
\[ \alpha = 0.042% = 0.00042 \]
Since the alternative hypothesis is two-tailed, we need to divide the significance level by 2 to find the critical values for each tail of the distribution:
\[ \alpha_{\text{one-tailed}} = \frac{0.00042}{2} = 0.00021 \]
Next, we will find the critical z-scores that correspond to the lower and upper tails. We can use the standard normal distribution table or a calculator with statistical functions to find these critical values.
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For the lower tail, we find the z-score corresponding to \(0.00021\): \[ P(Z < z) = 0.00021 \] Looking this up or using a calculator, we find: \[ z \approx -3.80 \]
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For the upper tail, we calculate the z-score corresponding to \(1 - 0.00021\): \[ P(Z < z) = 1 - 0.00021 = 0.99979 \] Again, looking this up or using a calculator, we find: \[ z \approx 3.80 \]
Critical Values:
Thus, the critical values for a two-tailed test at the 0.042% significance level are approximately:
\[ z = -3.8 \quad \text{and} \quad z = 3.8 \]
In conclusion, if the test statistic falls outside the range \([-3.8, 3.8]\), we would reject the null hypothesis in favor of the alternative hypothesis \(p \neq 0.7\).
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