Question
Assume that the normal distribution applies and find the critical values.
1) The significance level is 0.042% and alternative hypothesis is: p≠0.7.
1) The significance level is 0.042% and alternative hypothesis is: p≠0.7.
Answers
GPT-4o mini
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.
1. 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:
\[
z \approx -3.80
\]
2. 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 \(p \neq 0.7\).
\[
\alpha = 0.042\% = 0.00042
\]
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.
1. 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:
\[
z \approx -3.80
\]
2. 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 \(p \neq 0.7\).