Final answer:
To calculate the lower bound of a 95% confidence interval for the proportion of residents opposed to red light cameras, the sample proportion is determined to be 0.6571. The margin of error is calculated using the Z-score for a 95% confidence level, resulting in a lower bound of approximately 0.6217 or 62.17%.
Step-by-step explanation:
To find the lower bound of a 95% confidence interval for the population proportion in this survey, where 460 out of 700 residents are opposed to the use of red light cameras, we use the formula for the confidence interval of a population proportion. The formula involves calculating the sample proportion (p'), adding and subtracting the margin of error (ME) from it:
- p' = x/n = 460/700 = 0.6571 (sample proportion)
- n = 700 (sample size)
Next, to find ME, we use the following formula:
- ME = Z * sqrt[(p'(1-p')/n)]
- For a 95% confidence interval, Z is typically 1.96.
Now we calculate ME:
- ME = 1.96 * sqrt[(0.6571(1-0.6571)/700)]
- ME ≈ 0.0354
Finally, we calculate the lower bound:
- Lower Bound = p' - ME = 0.6571 - 0.0354 = 0.6217
The lower bound of the confidence interval is thus approximately 0.6217, or 62.17%. This means we are 95% confident that the true proportion of the population opposed to red light cameras is at least 62.17%.