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A confidence interval gives us a range in which we expect the true value of a population parameter to fall, based on our sample data. When we talk about a 90% confidence interval, we mean that if we were to take 100 different samples and compute a confidence interval in the same way for each sample, we would expect the true population parameter to fall within these intervals in 90 out of the 100 samples. For the problem at hand, the environmental agency wants to be 90% confident in their estimate of faulty car emissions.

The confidence level directly influences the accuracy of our estimate. Higher confidence levels will result in wider intervals, implying more uncertainty. Conversely, a lower confidence level will yield narrower intervals, indicating more precision but less certainty.

The margin of error tells us how much we expect our sample estimates to differ from the actual population parameter. In this exercise, the agency sets a margin of error at 3%. This means they want to ensure their estimate is no more than 3% above or below the true proportion of non-compliant vehicles.

To ensure accuracy within this margin, the sample size must be large enough to account for natural variations in the data. The smaller the margin of error, the larger the sample size required because we want our estimate to be very close to the true, unknown value.

Proportion estimation is used when we want to infer about a population's characteristic based on sample data. In this context, we estimate what fraction of cars do not meet emission standards.

We calculate the sample proportion as \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of faulty cars and \( n \) is the total size of the sample. From the initial pilot study, the sample proportion is 0.15, meaning 15% of the cars are faulty in the tested group. The sample proportion serves as an estimate of the true population proportion, guiding our sample size calculations.

The Z-score is a measure of how many standard deviations an element is from the mean. It also helps us determine the confidence level. For a 90% confidence level, the corresponding Z-score is 1.645. This Z-score shows that the interval around the sample proportion should cover 90% of the distribution.

Z-scores can be found in a standard normal distribution table or using statistical software. In sample size calculation, they play a critical role by fixing how wide the confidence interval should be, based on how much uncertainty we're willing to tolerate. By leveraging Z-scores, we can compute precise sample sizes that balance confidence and specificity.