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When dealing with proportions in statistics, the sample proportion is a foundation concept. It is essentially the fraction of the sample that exhibits a particular trait or outcome. In this exercise, we are interested in how many people out of a surveyed group watch educational television.

The formula to compute the sample proportion, denoted as \( \hat{p} \), is:

• \( \hat{p} = \frac{x}{n} \)

Here, \( x \) represents the number of people who watch educational TV, and \( n \) stands for the total number of people in the sample. By applying this formula with our numbers, \( x = 154 \) and \( n = 200 \), we find that \( \hat{p} = 0.77 \).

This means that 77% of the sampled individuals are viewers of educational television, providing us with initial insight into viewer habits in this population.

The Z-score is a critical value derived from the standard normal distribution, and it enables us to estimate the confidence interval of our sample proportion.

For a confidence interval, the Z-score helps determine how "confident" you are that a true parameter lies within your calculated interval. In the context of a 90% confidence interval, the Z-score represents the point where central 90% of our data lies within. From statistical tables, the Z-score for 90% confidence is approximately 1.645.

This specific score allows us to balance precision with the range, ensuring that our estimate for the television viewership proportion is as accurate as possible without being overly narrow.

Margin of Error (ME) represents the range around our sample proportion within which we expect the true population parameter to fall. It acts as a buffer for the interval.

The margin of error is calculated by multiplying the Z-score by the Standard Error (SE) of the proportion. The formula appears as follows:

• \( ME = Z \times SE \)

For our calculations, the Margin of Error is found by multiplying the Z-score 1.645 by the SE, approximately 0.0291, resulting in a ME of roughly 0.0479.

This implies that our estimate of the sample proportion could deviate by about 4.79% above or below, explaining the possible variability in sampling.

Standard Error (SE) quantifies the variation of your sample proportion owing to sampling variability. It is crucial for constructing confidence intervals.

The standard error is calculated using this formula:

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

This formula integrates our sample proportion's variance by multiplying \( \hat{p} \) and \( 1-\hat{p} \), then dividing by the sample size \( n \). For our scenario, substituting \( \hat{p} \) with 0.77 and \( n \) with 200 yields SE approximately 0.0291.

This small standard error signifies high precision in our sampling process, reinforcing the reliability of the population proportion estimation.