Question
Assume that the normal distribution applies and find the critical values.
The significance level is 0.00048 and a left tailed test statistic is used.
The significance level is 0.00048 and a left tailed test statistic is used.
Answers
GPT-4o mini
To find the critical value for a left-tailed test with a significance level (\(\alpha\)) of 0.00048 using the normal distribution, you can use a standard normal distribution table (Z-table) or a calculator that provides percentiles for the normal distribution.
1. **Identify the significance level:** Our significance level (\(\alpha\)) is 0.00048. This represents the probability of obtaining a test statistic that falls in the left tail of the distribution.
2. **Find the critical value:** Since this is a left-tailed test, we are looking for the Z-score that corresponds to the cumulative probability of \(1 - \alpha\). This means we need to find the Z-score for:
\[
P(Z < z) = \alpha = 0.00048
\]
3. **Using the Z-table or calculator:** You will look up or calculate the Z-score that corresponds to the cumulative probability of 0.00048.
Using a Z-table or normal distribution calculator:
- You look for the Z value such that the area to the left is 0.00048.
The Z-score corresponding to \(P(Z < z) = 0.00048\) is approximately -3.11.
4. **Conclusion:** The critical value at the significance level of 0.00048 for a left-tailed test is approximately:
\[
z \approx -3.11
\]
This means if your test statistic is less than -3.11, you would reject the null hypothesis.
1. **Identify the significance level:** Our significance level (\(\alpha\)) is 0.00048. This represents the probability of obtaining a test statistic that falls in the left tail of the distribution.
2. **Find the critical value:** Since this is a left-tailed test, we are looking for the Z-score that corresponds to the cumulative probability of \(1 - \alpha\). This means we need to find the Z-score for:
\[
P(Z < z) = \alpha = 0.00048
\]
3. **Using the Z-table or calculator:** You will look up or calculate the Z-score that corresponds to the cumulative probability of 0.00048.
Using a Z-table or normal distribution calculator:
- You look for the Z value such that the area to the left is 0.00048.
The Z-score corresponding to \(P(Z < z) = 0.00048\) is approximately -3.11.
4. **Conclusion:** The critical value at the significance level of 0.00048 for a left-tailed test is approximately:
\[
z \approx -3.11
\]
This means if your test statistic is less than -3.11, you would reject the null hypothesis.