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Understanding the concept of sample proportion is essential for constructing confidence intervals. The sample proportion, denoted as \(\text{\hat{p}}\), represents the fraction of the sample that exhibits a certain characteristic or success. It is computed using the formula:
\[ \text{\hat{p}} = \frac{x}{n} \] where \(x\) is the number of successes in the sample, and \(n\) is the total sample size.
In our example, there are 540 successes out of 900 total observations, so the sample proportion is calculated as \[ \text{\hat{p}} = \frac{540}{900} = 0.6 \. \]
This tells us that 60% of the sample exhibits the characteristic of interest.

Next, let's delve into the standard error (SE) which is a crucial component in confidence interval calculations. The standard error provides a measure of how much the sample proportion is expected to vary from the true population proportion due to random sampling. The formula to compute standard error is:
\[ \text{SE} = \sqrt{\frac{\text{\hat{p}} (1 - \text{\hat{p}})}{n}} \. \]
Substituting in our values, with \( \text{\hat{p}} = 0.6 \) and \( n = 900 \), we get:
\[ \text{SE} = \sqrt{\frac{0.6 \times 0.4}{900}} = 0.01633 \. \]
So, the standard error here is approximately 0.01633. This helps us understand the accuracy of our sample proportion in estimating the true population proportion.

Z-score, also known as the standard normal deviate, is another important concept. It represents the number of standard deviations a data point is from the mean in a standard normal distribution. For confidence interval calculations, the Z-score is used to determine the critical value corresponding to the desired confidence level.
Common confidence levels and their corresponding Z-scores include:

• 90%: Z = 1.645
• 95%: Z = 1.96
• 96%: Z = 2.05
• 99%: Z = 2.576

In our example, for a 96% confidence level, the Z-score is 2.05. This means that in a normal distribution, 96% of the area under the curve lies within 2.05 standard deviations of the mean.

The margin of error (ME) combines the Z-score and the standard error to provide a range within which we can expect the true population proportion to lie. It is calculated using the formula:
\[ \text{ME} = Z \times \text{SE} \. \]
Using our values, \( Z = 2.05 \) and \( \text{SE} = 0.01633 \, \) we get:
\[ \text{ME} = 2.05 \times 0.01633 = 0.0335 \. \]
So, the margin of error is approximately 0.0335. Finally, we construct the confidence interval by adding and subtracting the margin of error from the sample proportion:
The confidence interval is \[ 0.6 \pm 0.0335, \] or equivalently \[ 0.5665 \text{ to } 0.6335 \. \]
This interval suggests that we are 96% confident that the true population proportion lies between 0.5665 and 0.6335.