Final answer:
The 95% confidence interval for the proportion of automobiles in the state with emission levels exceeding the standard is approximately (0.143, 0.657), indicating with 95% confidence that the true proportion lies within this range.
Step-by-step explanation:
To find the 95% confidence interval for the proportion of automobiles in the state whose emission levels exceed the standard, we use the formula for the confidence interval of a proportion which is:
CI = p ± Z* √(p(1-p)/n)
Where:
- p is the sample proportion, which is 28/70 = 0.4
- n is the size of the sample, which is 70
- Z* is the Z-score corresponding to the desired confidence level, which is 1.96 for 95%
First calculate the standard error (SE):
SE = √(p(1-p)/n) = √(0.4(0.6)/70) = √(0.017143) ≈ 0.1309
Next, calculate the margin of error (ME):
ME = Z* × SE = 1.96 × 0.1309 ≈ 0.2566
Now, compute the lower and upper bounds of the confidence interval:
Lower bound = p - ME = 0.4 - 0.2566 ≈ 0.1434
Upper bound = p + ME = 0.4 + 0.2566 ≈ 0.6566
Therefore, the 95% confidence interval for the proportion is approximately (0.143, 0.657), rounded to three decimal places.
The interval means we are 95% confident that the true proportion of automobiles with emission levels above the standard in the state falls between 14.3% and 65.7%.