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A sample proportion is a basic concept in statistics. It represents the fraction or percentage of a sample with a particular attribute or characteristic of interest. In the context of a survey, it is often denoted as \( \hat{p} \). This value is used to estimate the true proportion for the entire population.
For instance, in the given exercise, a marketing company surveyed 200 households. They found that in 120 of them, women predominantly made purchasing decisions. Thus, the sample proportion \( \hat{p} \) is calculated as \( \hat{p} = \frac{120}{200} = 0.6 \). This means that within the sample, 60% of households observed exhibit the characteristic of interest.
Understanding the sample proportion serves as the foundation for constructing a confidence interval, aiding in the broader estimation of the true population proportion.

The standard error (SE) is a statistical term that measures the accuracy with which a sample represents a population. When estimating the true population proportion based on a sample, the standard error quantifies the expected variation or 'spread' of the sample proportion.
Mathematically, the standard error of the sample proportion is determined using the formula:

• \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

where \( \hat{p} \) is the sample proportion and \( n \) is the sample size. In this exercise, with \( \hat{p} = 0.6 \) and a sample size \( n = 200 \), the standard error is calculated as:

• \[ SE = \sqrt{\frac{0.6 \times (1 - 0.6)}{200}} = \sqrt{\frac{0.24}{200}} = 0.03464 \]

This value tells us how much the sample proportion is expected to fluctuate from the actual population proportion, assuming that our sample is representative.

A Z-score is a statistical measurement that describes how many standard deviations a data point is from the mean of a set of data. In the context of confidence intervals, the Z-score (also referred to as the critical value) is crucial in determining the range in which the true population parameter is likely to fall.
For a given confidence level, we determine a corresponding Z-score. For example, a 95% confidence interval is commonly used, and its Z-score is approximately 1.96. This means that in a standard normal distribution, 95% of the data falls within 1.96 standard deviations from the mean.
This Z-score is used along with the standard error to find the error bound, a key step in calculating confidence intervals.

The error bound in a confidence interval is what allows us to understand the precision of our estimate. It represents the maximum expected difference between the sample proportion and the true population proportion, given a specific confidence level.
To compute the error bound \( E \) for a proportion, the following formula is used:

• \[ E = Z \times SE \]

where \( Z \) is the Z-score for our confidence level, and \( SE \) is the standard error. From the exercise, for a 95% confidence level, we use a Z-score of 1.96. The previously calculated standard error is 0.03464. Plugging these values into the formula gives us:

• \[ E = 1.96 \times 0.03464 \approx 0.0678 \]

This means the true proportion is within \( 0.0678 \) of the sample proportion, on either side, with 95% confidence.
The error bound is added and subtracted from the sample proportion to create the confidence interval, ensuring a larger picture of the population proportion estimate.