Final answer:
To calculate a 95% confidence interval for the surveyed drivers' opinions with a skewed distribution, we apply the Central Limit Theorem since the sample size is large. With a mean of 2.12 and standard deviation of 1.65, the interval is approximately (2.02, 2.22), and skewness should not strongly affect the interval.
Step-by-step explanation:
To calculate a 95% confidence interval for the mean opinion of all licensed drivers based on a survey of 958 responses with a sample mean of x = 2.12 and a standard deviation of σ = 1.65, we can use the formula for a confidence interval when the population standard deviation is known:
Confidence interval = x ± Z*(σ/√n)
Where Z* is the Z-score corresponding to a 95% confidence level, which is typically 1.96 for a normal distribution. However, since the distribution is skewed and not normal, this might slightly affect the accuracy of the Z-score, but the Central Limit Theorem states that for large samples (n > 30), the distribution of the sample means will be approximately normal regardless of the shape of the population distribution.
Thus, we calculate:
Error Margin (EM) = Z*(σ/√n) = 1.96*(1.65/√958) = 1.96*0.0533 = 0.1045 (rounded to two decimal places is 0.10)
The 95% confidence interval is therefore:
2.12 ± 0.10 = (2.02, 2.22)
This interval is with ±0.01 precision. Although the underlying distribution is skewed, with a large sample size such as 958, the Central Limit Theorem assures us that the sample mean will be normally distributed, and thus the skewness does not strongy affect the confidence interval.