Math Problem Statement
Solution
Let's go through this problem step-by-step.
Given Information:
- Test Statistic:
- Claim: (indicating a right-tailed test)
- Significance Level:
Part (a): Finding the Critical Value(s)
Since this is a right-tailed test with a 0.01 significance level, we need to find the critical value that corresponds to the upper 1% of the standard normal distribution.
Using the standard normal table or Z-scores, we find that:
- The critical value for a right-tailed test at is approximately 2.33.
So the answer for part (a) is:
- Option A: There is one critical value; the critical value is 2.33.
Part (b): Should we reject or fail to reject ?
The test statistic is compared to the critical value of 2.33:
- Since , the test statistic falls in the critical region for a right-tailed test.
Therefore, we should reject .
The answer for part (b) is:
- Option B: should be rejected since the test statistic is in the critical region.
Final Answers:
- (a) Option A: The critical value is 2.33.
- (b) Option B: Reject .
Would you like further clarification on any step, or do you have any other questions?
Additional Questions for Exploration:
- What would be the critical value if the significance level was 0.05 instead of 0.01?
- How would the decision change if the test statistic was 2.00 instead of 2.83?
- Why does the direction of the test (right-tailed vs. left-tailed) affect the critical region?
- What is the interpretation of rejecting in this context?
- How would you find the critical value for a two-tailed test with the same significance level?
Tip:
For hypothesis testing, always check the direction of the test (right-tailed, left-tailed, or two-tailed) to correctly identify the critical value and interpret the test statistic.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Z-Test
Critical Value
Significance Level
Formulas
Z-test formula
Critical value determination for one-tailed test
Theorems
Standard Normal Distribution
Critical Region for Hypothesis Testing
Suitable Grade Level
College Level (Statistics)
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