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The sample mean is a simple but powerful concept in statistics. It represents the average value of a set of numerical data collected from a sample. In our exercise, the Gallup Poll surveyed 398 adult women, and their average weight, the sample mean, was calculated to be 155 pounds.
This value is used as an estimate of the true mean of the population, meaning the average weight of all adult women we want to understand. Good sampling is crucial because it allows this estimate to be as close to the true population mean as possible. The reason behind using the sample mean is that in many cases, getting data from the entire population is not feasible due to time or resource constraints.

The standard error (SE) is a measure that helps us understand the precision of the sample mean as an estimate of the population mean. It tells us how much the sample mean is expected to vary from one sample to another. Smaller SE indicates more reliable estimates.
In our exercise, the formula for the SE is \[ SE = \frac{\sigma}{\sqrt{n}} \]where \(\sigma\) is the population standard deviation, and \(n\) is the sample size. Using the provided values, \[ SE = \frac{35}{\sqrt{398}} \approx 1.755 \].
This SE shows that even though we only sampled a group of women, the estimate of the average weight is within approximately 1.755 pounds of the actual average weight of the entire population.

A Z-score is a statistical measure that describes a value's relation to the mean of a group of values. It indicates how many standard deviations an element is from the mean. In the context of confidence intervals, the Z-score determines the "spread" of data around the sample mean.
To construct a 95% confidence interval for the mean, a Z-score of 1.96 is used. This value comes from the standard normal distribution and corresponds to 0.025 probability in each tail (2.5% on either extreme), ensuring that we cover 95% of the possible sample means. The Z-score helps in establishing how wide the confidence interval should be, which directly impacts how confident we are that the interval contains the true population mean.

The margin of error (MOE) is a crucial component in estimating how accurate the sample mean is likely to be. It defines the range in which we expect the true population mean to fall, given our sample data.
The MOE is calculated by multiplying the Z-score by the standard error: \[ MOE = Z \times SE \]. Using the values from the exercise, \[ MOE = 1.96 \times 1.755 \approx 3.4438 \].
This means the true mean weight of all adult women would most likely fall within the interval created by adding and subtracting this margin from the sample mean. In our example, the confidence interval is \[ 155 \pm 3.4438 \], resulting in approximately \[ (151.5562, 158.4438) \]. This interval suggests that if we were to take many samples, about 95% of those intervals would capture the true mean, reflecting the inherent variability in sample estimates.