Math Problem Statement
Determine whether you would reject or fail to reject the null hypothesis in the following situations: t = 2.58, N = 21, two-tailed test a = 0.05
Solution
To determine whether we reject or fail to reject the null hypothesis, we'll follow the steps for hypothesis testing using a two-tailed t-test.
Given:
- (sample size)
- Two-tailed test
- (significance level)
Step-by-step solution:
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Degrees of freedom (df): The degrees of freedom for the t-distribution is 20.
-
Critical value of t: For a two-tailed test with , we need to split the significance level between both tails, so each tail will have .
Using a t-distribution table or a calculator, the critical value of t for and (two-tailed) is approximately:
-
Decision rule: If the calculated t-statistic falls outside the range of the critical values (), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
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Comparison: The calculated -value is .
Since , it falls in the rejection region of the t-distribution.
Conclusion:
Since the calculated t-statistic is greater than the critical value , we reject the null hypothesis at the significance level.
Would you like to go over more details or have any specific questions about this process?
Here are 5 related questions that expand on this topic:
- How do you calculate the p-value for this t-test?
- What would happen if we changed to 0.01 instead of 0.05?
- How do you conduct a one-tailed t-test, and how would the critical value change?
- Why is it important to know if the test is one-tailed or two-tailed?
- How does increasing the sample size affect the critical value and decision?
Tip: Always ensure you know whether you're working with a one-tailed or two-tailed test, as it affects the critical value and interpretation!
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Math Problem Analysis
Mathematical Concepts
Statistics
Hypothesis Testing
t-distribution
Formulas
Degrees of Freedom: df = N - 1
Critical t-value for two-tailed test at α = 0.05
Theorems
Central Limit Theorem
t-test hypothesis testing
Suitable Grade Level
College-level Statistics
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