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Calculating the sample proportion is the first crucial step in constructing a confidence interval for a population proportion. The sample proportion, denoted as \( \hat{p} \), is simply the ratio of successful events (\( x \)) to the total sample size (\( n \)). In mathematical terms, this is represented as follows: \[ \hat{p} = \frac{x}{n} \] For instance, in our given exercise, we have 30 successful cases out of a total sample size of 150. Plugging these values into the formula gives us: \[ \hat{p} = \frac{30}{150} = 0.2 \] This sample proportion of 0.2 indicates that 20% of the sample has the attribute of interest. This proportion will form the baseline for further calculations.

The critical value is a key component in the construction of confidence intervals. It represents the number of standard deviations you must go out from the mean to capture a given percentage of the population distribution. For a 90% confidence level, this critical value (denoted as \( z^* \)) corresponds to the point where the cumulative probability is 0.95 (since 90% confidence leaves 5% equally split in the two tails). This value is commonly found using Z-tables or standard normal distribution tables. For our purpose, the critical value for a 90% confidence interval is: \[ z^* = 1.645 \] This means that, around the mean proportion, we have to span out 1.645 standard deviations in both directions to achieve the desired confidence level.

The standard error (SE) of the sample proportion quantifies the amount of variability or dispersion we can expect when estimating the population proportion from a sample. It is calculated using this formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] In our exercise, with a sample proportion of \( \hat{p} = 0.2 \) and a sample size of \( n = 150 \), the standard error is computed as: \[ SE = \sqrt{\frac{0.2(1 - 0.2)}{150}} = \sqrt{\frac{0.16}{150}} = \sqrt{0.0010667} \approx 0.032666 \] Hence, the standard error here is approximately 0.032666, indicating the expected variability in the sample proportion estimate.

The margin of error (ME) provides a range within which we believe the true population proportion lies, given our sample data. It is calculated by multiplying the critical value (\( z^* \)) by the standard error (SE): \[ ME = z^* \times SE \] Using our critical value of 1.645 and the standard error of approximately 0.032666, we get: \[ ME = 1.645 \times 0.032666 \approx 0.0537 \] This means the margin of error is around 0.0537. It reflects the degree of uncertainty around our estimate of the sample proportion.

Finally, constructing the confidence interval involves combining our sample proportion with the margin of error. The confidence interval is formulated as follows: \[ (\hat{p} - ME, \hat{p} + ME) \] For our problem, this becomes: \[ (0.2 - 0.0537, 0.2 + 0.0537) \] Therefore, the 90% confidence interval for the population proportion is: \[ (0.1463, 0.2537) \] This interval means we are 90% confident that the true population proportion lies between 0.1463 and 0.2537. Confidence intervals help us understand the range within which the actual population parameter is likely to lie, increasing our estimation reliability.