Question 1168845: A survey of licensed drivers inquired about running red lights. One question asked, "Of every 10 motorists who run a red light, about how many do you think will be caught?" The mean result for 880 respondents was
¯x= 1.92. Suppose we know that
σ= 1.83. Compute a 95% confidence interval for the mean opinion in the population of all licensed drivers. (Round your answer to the nearest hundredth.)
(,)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's calculate the 95% confidence interval for the mean opinion.
**Given Information:**
* Sample mean (x̄) = 1.92
* Population standard deviation (σ) = 1.83
* Sample size (n) = 880
* Confidence level = 95%
**Steps:**
1. **Find the Z-score for the Confidence Level:**
* For a 95% confidence interval, the z-score is 1.96. This value corresponds to the z-scores that capture 95% of the area under the standard normal distribution.
2. **Calculate the Standard Error:**
* Standard error (SE) = σ / √n
* SE = 1.83 / √880
* SE ≈ 1.83 / 29.66479
* SE ≈ 0.06168
3. **Calculate the Margin of Error:**
* Margin of error (ME) = z * SE
* ME = 1.96 * 0.06168
* ME ≈ 0.12089
4. **Calculate the Confidence Interval:**
* Confidence interval = x̄ ± ME
* Lower bound = x̄ - ME = 1.92 - 0.12089 ≈ 1.79911
* Upper bound = x̄ + ME = 1.92 + 0.12089 ≈ 2.04089
5. **Round to the Nearest Hundredth:**
* Lower bound ≈ 1.80
* Upper bound ≈ 2.04
**Therefore, the 95% confidence interval for the mean opinion in the population of all licensed drivers is (1.80, 2.04).**
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