Final answer:
To calculate the 90% confidence interval for the proportion of company employees who carpool, the sample proportion and size are used along with the normal distribution's z-score, resulting in a confidence interval of approximately (0.53%, 2.77%), with the closest answer option being D.
Step-by-step explanation:
To find the 90% confidence interval for the percentage of all employees of the company who carpool based on 243 employees with 1.65% carpooling, we can use the one-proportion z-interval procedure. First, we need to identify the sample proportion (phat) and the sample size (n).
Sample proportion phat = 1.65% = 0.0165
Sample size n = 243
Next, we find the standard error of the proportion using the formula SE = √(phat * (1 - phat) / n).
SE = √(0.0165 * (1 - 0.0165) / 243) ≈ 0.0068
To find the z-score for a 90% confidence level, we look up the z-value that cuts off the upper 5% of the normal curve, which is about 1.645 (since we leave out 5% on each tail for a 90% confidence interval).
Now we can calculate the margins of error: ME = z-score * SE = 1.645 * 0.0068 ≈ 0.0112. Finally, we add and subtract the margin of error from our sample proportion to find the confidence interval:
CI = phat ± ME = 0.0165 ± 0.0112 = (0.0053, 0.0277)
Converting back to percentages: CI = (0.53%, 2.77%). The correct answer is close to option D, which is 0.306% to 2.994%.