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The population mean is a measure that indicates the average outcome for a certain population. When we collect data from a sample, like the number of children women have in the exercise, we are aiming to make inferences about the entire population’s mean value. Since it's impractical to survey every individual in a population, a sample provides a manageable and cost-effective approximation.

For instance, if a survey found that the average number of children is 1.85 for a sample of women, this sample mean is used as an estimate for the population mean. The population mean is a key element in calculating confidence intervals as it represents the value we are trying to estimate and make conclusions about.

The standard error is a statistic that measures the precision of the sample mean as an estimate of the population mean. Essentially, it reflects the variability of the sample means you might get if you took multiple samples from the same population.

In the context of our exercise, we calculate the standard error by dividing the sample standard deviation (1.5) by the square root of the sample size (\(sqrt{100}\)). This calculation results in a standard error of 0.15, which we use to understand how much the sample mean (1.85) might fluctuate around the population mean. A smaller standard error indicates a more precise estimate, which ultimately affects the width of the confidence interval.

A critical value is a point on the distribution curve that separates the area where the null hypothesis is accepted from the area where the null hypothesis is rejected. It is contingent on the desired level of confidence and the distribution being used.

In the given exercise, the critical values 1.96 and 1.645 are used for calculating the 95% and 90% confidence intervals, respectively. These values represent the Z-scores for the standard normal distribution. To clarify, these Z-scores symbolize how many standard deviations away from the mean a certain point is. The larger the critical value, the wider the confidence interval, as we’ve observed in the step-by-step solution.

Sample size plays a pivotal role in statistical analysis. In essence, the larger the sample, the closer you are likely to get to the true mean of the population, hence, the lower your standard error and the more precise your estimate will be.

In our example, a sample size of 100 is large enough to apply the central limit theorem and justify the use of Z-scores instead of T-scores for constructing confidence intervals. It's also why the sample mean of 1.85 children is likely a strong estimate of the true population mean. Nevertheless, larger sample sizes would further refine our confidence interval, making it narrower and providing a more precise range for the population mean.

Statistical significance indicates the likelihood that the result from a study or experiment is due to something other than random chance. It's a measure of the strength of evidence when testing hypotheses.

The confidence levels used in our exercise (90% and 95%) relate directly to statistical significance. They indicate the degree to which we can be confident that the population parameter (mean number of children) will fall within the estimated interval. As the confidence level increases, we require more evidence to reject the null hypothesis and, therefore, the confidence interval becomes wider. It flags the trade-off between certainty and precision when interpreting statistical results.