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The sample proportion, often denoted by \( \hat{p} \), is a way of expressing the portion of a group exhibiting a particular trait. In the example provided, \( 48\% \) of 331 Americans surveyed said they do not attend college because they cannot afford it. This means the sample proportion \( \hat{p} \) is \( 0.48 \).

To get this value:

• Translate the percentage into a proportion by dividing by 100.
• In this case, \( \hat{p} = 0.48 \), reflecting 48 out of 100 chose not to attend college for financial reasons.

This measure is crucial, as it forms the basis of further statistical calculations, like the standard error and confidence intervals. Understanding the sample proportion helps in estimating the same behavior in the overall population.

The standard error (SE) of a sample proportion provides an estimation of the variability of the sample proportion from the true population proportion. It’s calculated using the formula: \[ SE = \sqrt{\frac{\hat{p} (1-\hat{p})}{n}} \] where \( \hat{p} = 0.48 \) and \( n = 331 \) in this context.

Let’s break this down:

• \( \hat{p} \) is the sample proportion.
• \( 1 - \hat{p} = 0.52 \) represents the proportion of those who do not fall into the observed category.
• \( n \) is the sample size.

Plug these numbers into the formula to get the SE as approximately \( 0.0274 \).

This SE value indicates the degree of variation expected in the sample proportion, helping inform the range within which the true population proportion likely falls.

The margin of error (ME) in a confidence interval indicates the extent of possible error in the estimation of the population proportion. It is computed by multiplying the standard error by the critical value \( z^* \) corresponding to the desired confidence level. For a 90% confidence level, \( z^* \approx 1.645 \).

The formula used is: \[ ME = z^* \times SE \]
In our example, it becomes: \( ME \approx 1.645 \times 0.0274 \approx 0.0451 \).

This means plus or minus approximately 4.51%.

• The confidence interval is established as \( \hat{p} \pm ME \).
• Calculating this gives a range of \([0.4349, 0.5251]\).

This range implies that, with 90% confidence, the true proportion of Americans not affording college lies within these bounds.

Determining the required sample size is essential for achieving a desired margin of error in surveys or experiments. If a specific margin of error is targeted, the formula to calculate the needed sample size \( n \) is rearranged from the margin of error equation: \[ n = \left(\frac{z^*}{ME}\right)^2 \times \hat{p} (1-\hat{p}) \]

This formula allows you to solve for \( n \) given \( z^* \), \( ME \), and \( \hat{p} \). In the exercise's scenario:

• Use \( z^* = 1.645 \) for 90% confidence.
• Substitute \( ME = 0.015 \) for a 1.5% error margin.
• The previous \( \hat{p} = 0.48 \).

Plugging in values, the calculation gives: \( n \approx \left(\frac{1.645}{0.015}\right)^2 \times 0.48 \times 0.52 \approx 4600 \).

This indicates that to achieve the tighter margin of error, around 4600 respondents are needed, ensuring reliable and confidence-backed results in reflecting the population's views on college affordability.