SOLUTION: For the following hypothesis test: H0: μ ≤ 70 HA: μ > 70 with n = 20, x = 71.2, s = 6.9, and a = 0.1, state a. the decision rule in terms of the critical value of the test

Question 1175530: For the following hypothesis test:
H0: μ ≤ 70
HA: μ > 70
with n = 20, x = 71.2, s = 6.9, and a = 0.1, state
a. the decision rule in terms of the critical value of the test statistic [2]
b. the calculated value of the test statistic [2]
c. the conclusion

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's break down this hypothesis test step-by-step.
**a) Decision Rule in Terms of the Critical Value**
* **Test Type:** This is a right-tailed t-test because the alternative hypothesis (HA: μ > 70) indicates we're looking for evidence that the mean is greater than 70.
* **Degrees of Freedom:** The degrees of freedom (df) are n - 1 = 20 - 1 = 19.
* **Significance Level:** The significance level (alpha) is 0.1.
* **Critical Value:** We need to find the critical t-value (t_critical) that corresponds to an alpha of 0.1 and 19 degrees of freedom. Using a t-table or calculator, we find that t_critical ≈ 1.328.
* **Decision Rule:** Reject the null hypothesis (H0) if the calculated t-statistic is greater than 1.328.
**b) Calculated Value of the Test Statistic**
* **Formula:** The t-statistic is calculated using the formula:
* t = (x̄ - μ) / (s / √n)
* where:
* x̄ is the sample mean (71.2)
* μ is the hypothesized population mean (70)
* s is the sample standard deviation (6.9)
* n is the sample size (20)
* **Calculation:**
* t = (71.2 - 70) / (6.9 / √20)
* t = 1.2 / (6.9 / 4.472)
* t = 1.2 / 1.543
* t ≈ 0.778
**c) Conclusion**
* **Comparison:** We compare the calculated t-statistic (0.778) to the critical t-value (1.328).
* **Decision:** Since 0.778 is less than 1.328, we fail to reject the null hypothesis.
* **Conclusion:** There is insufficient evidence to support the claim that the population mean is greater than 70 at the 0.1 significance level.

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