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:

  • t=2.58t = 2.58
  • N=21N = 21 (sample size)
  • Two-tailed test
  • α=0.05\alpha = 0.05 (significance level)

Step-by-step solution:

  1. Degrees of freedom (df): df=N1=211=20df = N - 1 = 21 - 1 = 20 The degrees of freedom for the t-distribution is 20.

  2. Critical value of t: For a two-tailed test with α=0.05\alpha = 0.05, we need to split the significance level between both tails, so each tail will have 0.0250.025.

    Using a t-distribution table or a calculator, the critical value of t for df=20df = 20 and α=0.05\alpha = 0.05 (two-tailed) is approximately: tcritical±2.086t_{\text{critical}} \approx \pm 2.086

  3. Decision rule: If the calculated t-statistic falls outside the range of the critical values (tcritical=±2.086t_{\text{critical}} = \pm 2.086), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

  4. Comparison: The calculated tt-value is t=2.58t = 2.58.

    Since 2.58>2.0862.58 > 2.086, it falls in the rejection region of the t-distribution.

Conclusion:

Since the calculated t-statistic 2.582.58 is greater than the critical value 2.0862.086, we reject the null hypothesis at the α=0.05\alpha = 0.05 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:

  1. How do you calculate the p-value for this t-test?
  2. What would happen if we changed α\alpha to 0.01 instead of 0.05?
  3. How do you conduct a one-tailed t-test, and how would the critical value change?
  4. Why is it important to know if the test is one-tailed or two-tailed?
  5. How does increasing the sample size NN 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