Question
A test of Hv0 : u =52 versus Hv1 : u > 52 is performed using a significance level of a=0.05. The value of the test statistic is z=1.97.
Determine whether to reject Hv0.
Determine whether to reject Hv0.
Answers
GPT-4o mini
To determine whether to reject \( H_0: \mu = 52 \) in favor of \( H_1: \mu > 52 \) with a significance level of \( \alpha = 0.05 \) and a test statistic of \( z = 1.97 \), we need to compare the test statistic to the critical value for a one-tailed test at the 0.05 significance level.
1. **Find the critical z-value**: For a significance level of \( \alpha = 0.05 \) in a one-tailed test, the critical z-value (z-critical) can be found using z-tables or standard normal distribution calculators. The critical value for \( \alpha = 0.05 \) is approximately \( z = 1.645 \).
2. **Compare the test statistic to the critical z-value**:
- Test statistic \( z = 1.97 \)
- Critical z-value \( z_{\text{critical}} = 1.645 \)
3. **Decision Rule**:
- If \( z > z_{\text{critical}} \), we reject \( H_0 \).
- If \( z \leq z_{\text{critical}} \), we do not reject \( H_0 \).
4. **Applying the decision rule**:
- Since \( 1.97 > 1.645 \), we reject \( H_0 \).
Conclusion: Based on the test statistic \( z = 1.97 \) being greater than the critical value of \( 1.645 \), we reject the null hypothesis \( H_0 \). There is sufficient evidence to support the alternative hypothesis \( H_1: \mu > 52 \) at the \( 0.05 \) significance level.
1. **Find the critical z-value**: For a significance level of \( \alpha = 0.05 \) in a one-tailed test, the critical z-value (z-critical) can be found using z-tables or standard normal distribution calculators. The critical value for \( \alpha = 0.05 \) is approximately \( z = 1.645 \).
2. **Compare the test statistic to the critical z-value**:
- Test statistic \( z = 1.97 \)
- Critical z-value \( z_{\text{critical}} = 1.645 \)
3. **Decision Rule**:
- If \( z > z_{\text{critical}} \), we reject \( H_0 \).
- If \( z \leq z_{\text{critical}} \), we do not reject \( H_0 \).
4. **Applying the decision rule**:
- Since \( 1.97 > 1.645 \), we reject \( H_0 \).
Conclusion: Based on the test statistic \( z = 1.97 \) being greater than the critical value of \( 1.645 \), we reject the null hypothesis \( H_0 \). There is sufficient evidence to support the alternative hypothesis \( H_1: \mu > 52 \) at the \( 0.05 \) significance level.
If the true value of u is 52, is the result a Type I error, a Type II error, or a correct decision?
GPT-4o mini
In hypothesis testing, we define two types of errors:
- **Type I Error**: This occurs when we reject the null hypothesis \( H_0 \) when it is actually true.
- **Type II Error**: This occurs when we fail to reject the null hypothesis \( H_0 \) when it is false.
In your scenario:
- The null hypothesis is \( H_0: \mu = 52 \).
- The alternative hypothesis is \( H_1: \mu > 52 \).
- You rejected \( H_0 \) based on the test statistic \( z = 1.97 \).
If the true value of \( \mu \) is indeed 52, this means that the null hypothesis is true. Since you rejected \( H_0 \) when it is actually true, you have made a **Type I error**.
In summary, in this case, if the true value of \( \mu \) is 52 and you rejected \( H_0 \), the result is a **Type I error**.
- **Type I Error**: This occurs when we reject the null hypothesis \( H_0 \) when it is actually true.
- **Type II Error**: This occurs when we fail to reject the null hypothesis \( H_0 \) when it is false.
In your scenario:
- The null hypothesis is \( H_0: \mu = 52 \).
- The alternative hypothesis is \( H_1: \mu > 52 \).
- You rejected \( H_0 \) based on the test statistic \( z = 1.97 \).
If the true value of \( \mu \) is indeed 52, this means that the null hypothesis is true. Since you rejected \( H_0 \) when it is actually true, you have made a **Type I error**.
In summary, in this case, if the true value of \( \mu \) is 52 and you rejected \( H_0 \), the result is a **Type I error**.
If the true value of is u, 67 is the result a Type I error, a Type II error, or a correct decision?
GPT-4o mini
Let's analyze the situation based on the revised true value of \( \mu = 67 \) while considering the hypotheses:
- Null hypothesis: \( H_0: \mu = 52 \)
- Alternative hypothesis: \( H_1: \mu > 52 \)
You previously rejected \( H_0 \) based on the test statistic \( z = 1.97 \).
### Analysis:
1. **True Value**: The true value of \( \mu \) is stated to be 67.
2. **Test Outcome**: You rejected \( H_0 \).
### Categorizing the Decision:
- **Correct Decision**: This occurs if you reject \( H_0 \) (which suggests that \( \mu > 52 \)) and the true value also supports this (e.g., \( \mu = 67\)).
- **Type I Error**: This would have occurred if you rejected \( H_0 \) when it was true (i.e., if \( \mu = 52\)).
- **Type II Error**: This occurs if you fail to reject \( H_0 \) and the alternative hypothesis is true (e.g., if \( \mu > 52\)).
### Conclusion:
Since you rejected \( H_0 \) and the true value is \( \mu = 67 \), which is indeed greater than 52, this is a **correct decision**.
To summarize: If the true value of \( \mu \) is 67 and you've rejected \( H_0: \mu = 52 \), you have made a **correct decision**.
- Null hypothesis: \( H_0: \mu = 52 \)
- Alternative hypothesis: \( H_1: \mu > 52 \)
You previously rejected \( H_0 \) based on the test statistic \( z = 1.97 \).
### Analysis:
1. **True Value**: The true value of \( \mu \) is stated to be 67.
2. **Test Outcome**: You rejected \( H_0 \).
### Categorizing the Decision:
- **Correct Decision**: This occurs if you reject \( H_0 \) (which suggests that \( \mu > 52 \)) and the true value also supports this (e.g., \( \mu = 67\)).
- **Type I Error**: This would have occurred if you rejected \( H_0 \) when it was true (i.e., if \( \mu = 52\)).
- **Type II Error**: This occurs if you fail to reject \( H_0 \) and the alternative hypothesis is true (e.g., if \( \mu > 52\)).
### Conclusion:
Since you rejected \( H_0 \) and the true value is \( \mu = 67 \), which is indeed greater than 52, this is a **correct decision**.
To summarize: If the true value of \( \mu \) is 67 and you've rejected \( H_0: \mu = 52 \), you have made a **correct decision**.