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The sample proportion represents the fraction of individuals in the sample with a particular trait of interest. In our case, the trait is favoring fluoridation among Utah residents. To calculate this proportion, we convert the percentage given into a decimal form. For the Associated Press article stating that 65% of the surveyed residents favor fluoridation, we have a sample proportion (denoted as \( \hat{p} \) ) of \( 0.65 \). It's key to understand that \( \hat{p} \) is an estimate of the true population proportion (\( p \) ), based on the sampled individuals.

Recalling that our sample size is 150 residents, this number is high enough to ensure that our approach to construct a confidence interval is appropriate. With larger samples, the estimate of the sample proportion becomes more accurate, giving us stronger evidence about the true population proportion.

The standard error (SE) is a critical statistic for understanding how much we'd expect our sample proportion to vary if we were to take many samples of the same size from the population. It's calculated using the formula \[ SE = \sqrt{\hat{p} ( 1 - \hat{p} ) / n} \], where \( \hat{p} \) is the sample proportion and \( n \) is the sample size. For our exercise, this would be \[ SE = \sqrt{0.65 * ( 1 - 0.65 ) / 150 } \], yielding approximately \( 0.0391 \).

The smaller the standard error, the more confident we can be about our sample proportion representing the true population proportion. It's also worth noting that the standard error decreases with an increase in the sample size, following the law of large numbers.

A z-score is a measure of how many standard deviations a data point is from the mean. In the context of constructing confidence intervals, the z-score corresponds to the desired confidence level. The z-score tells us how far we need to go from the sample proportion in either direction to capture the true population proportion with a certain level of confidence.

For a 90% confidence level, we use a z-score of approximately \( 1.645 \), which indicates that our confidence interval will extend \( 1.645 \) standard errors above and below our sample proportion. Using z-scores in confidence interval calculations is appropriate when we have a large sample size and a population distribution that is approximately normal.

The process of making conclusions about population parameters based on sample statistics is known as statistical inference. The confidence interval we construct is a key tool in this process. With the previous calculations, we determined a 90% confidence interval for the true proportion of Utah residents who favor fluoridation to be between 58.6% and 71.4%. This range suggests with high certainty—or 90% confidence—that the actual proportion of all Utah residents in favor lies within this interval.

Since the lower bound of our interval (58.6%) exceeds 50%, we can infer that a majority of the total population likely supports fluoridation. Thus, even though our interval is an estimate derived from a sample, it enables us to deduce information about the entire population with a quantified level of sureness.