Question 1199549: A number of minor automobile accidents occur at various high-risk intersections in Teton County despite traffic lights. The Traffic Department claims that a modification in the type of light will reduce these accidents. The county commissioners have agreed to a proposed experiment. Eight intersections were chosen at random, and the lights at those intersections were modified. The numbers of minor accidents during a 6-month period before and after the modifications were:
After : 3 7 7 0 4 6 8 2
Before : 5 7 6 4 8 9 8 10
At the 0.01 significance level, is it reasonable to conclude that the modification reduced
the number of traffic accidents?
a. What are the null and alternate hypotheses?
b. Compute the test statistic.
c. Compute the p-value.
d. What is your decision regarding the null hypothesis?
e. Interpret the result.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a. Hypotheses**
* **Null Hypothesis (H0):** The modification in the type of traffic light did not reduce the number of minor accidents.
* Mathematically: μ_before ≤ μ_after
* **Alternative Hypothesis (H1):** The modification in the type of traffic light reduced the number of minor accidents.
* Mathematically: μ_before > μ_after
**b. Compute the Test Statistic**
* **Data:**
* After: 3, 7, 7, 0, 4, 6, 8, 2
* Before: 5, 7, 6, 4, 8, 9, 8, 10
* **Paired t-test:** Since we are comparing the same intersections before and after the modification, we use a paired t-test.
* **Calculate the differences:**
* 5 - 3 = 2
* 7 - 7 = 0
* 6 - 7 = -1
* 4 - 0 = 4
* 8 - 4 = 4
* 9 - 6 = 3
* 8 - 8 = 0
* 10 - 2 = 8
* **Calculate the mean and standard deviation of the differences:**
* Mean (d̄) = (2 + 0 - 1 + 4 + 4 + 3 + 0 + 8) / 8 = 2.625
* Standard deviation (s_d) = 2.878 (approximately)
* **Calculate the t-statistic:**
* t = (d̄ - 0) / (s_d / √n)
* t = (2.625 - 0) / (2.878 / √8)
* t ≈ 2.59
**c. Compute the p-value**
* **Degrees of freedom (df) = n - 1 = 8 - 1 = 7**
* **Using a t-distribution table or statistical software:** Find the p-value associated with t = 2.59 and df = 7.
* **p-value ≈ 0.018**
**d. Decision Regarding the Null Hypothesis**
* **Significance level (α) = 0.01**
* **Compare p-value to α:**
* p-value (0.018) > α (0.01)
* **Decision:** Since the p-value is greater than the significance level, we **fail to reject the null hypothesis**.
**e. Interpretation**
* **Conclusion:** At the 0.01 significance level, there is **not enough evidence** to conclude that the modification in the type of traffic light reduced the number of minor accidents.
* **Explanation:** The observed difference in the number of accidents could be due to chance.
**Note:**
* This analysis assumes that the differences in the number of accidents are normally distributed.
* A paired t-test is appropriate because we are comparing the same locations before and after the modification.
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