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.

Question

Determine whether to reject Hv0.

Answer 1

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.

Answer 2

If the true value of u is 52, is the result a Type I error, a Type II error, or a correct decision?

Answer 3

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.

Answer 4

If the true value of is u, 67 is the result a Type I error, a Type II error, or a correct decision?

Answer 5

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.