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To determine the right sample size for achieving a desired accuracy in statistics or surveys, you need to calculate how many participants or data points are necessary to achieve a particular margin of error with a certain level of confidence. This is crucial in ensuring that the results you obtain can be generalized to a larger population.
In the given exercise, for the margin of error to be about 1.5% at a 90% confidence level, the calculation involves working backward from the margin of error formula. The formula used is:

• \[ ME = z^* \times \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \]

Where:

• \( ME \) is the desired margin of error (0.015 for 1.5%)
• \( z^* \) is the critical value for the confidence level, which is about 1.645 for 90%
• \( \hat{p} \) is the estimated proportion, given as 0.48
• \( n \) is the sample size

Solving the equation helps to find the necessary sample size:\[ n = \frac{0.2496}{\left(\frac{0.015}{1.645}\right)^2} \approx 3076 \].
Simply put, when you need a larger sample size like 3076 respondents, it means you want to increase the accuracy and reliability of your data, reducing the margin of error to make the survey results more trustworthy.

The margin of error is a statistical term that describes the amount of random sampling error in a survey's results. It gives a range within which you expect the true value of the population parameter to fall. It is directly related to sample size; a larger sample size typically results in a smaller margin of error.
In this problem, the initial calculation of the confidence interval had a margin of error of approximately 0.045 or 4.5%. This was computed using the formula:

• \[ ME = z^* \times SE \]

This indicates how much the proportion could vary above or below the sample proportion (48% in this case). When we aim for a margin of error of just 1.5%, it signifies a need for more precision.
Reducing the margin of error allows researchers to provide tighter confidence intervals, implying more confidence in the data being closer to the reality. For practical purposes, if you want to inform policy or make business decisions based on survey data, ensuring a smaller margin of error is generally more beneficial. However, achieving a smaller margin typically requires more resources to deal with larger sample sizes.

The critical value is a factor used to compute the margin of error and confidence interval. It represents the number of standard deviations a point is from the mean of a standard normal distribution, which helps in determining how far you go from the sample estimate to build the interval around it.
The exercise deals with a 90% confidence interval, which corresponds to a critical value, denoted as \( z^* \). For most common confidence levels, this value can be found in statistical tables or calculated using statistical software. Here, the critical value for a 90% confidence interval is approximately 1.645.
How do we use it? The critical value is multiplied by the standard error to calculate the margin of error. It effectively sets the "width" of our confidence interval. Thus:

• The confidence interval is shaped by the formula \[ \hat{p} \pm z^* \times SE \]

Using this critical value allows us to claim, statistically speaking, that if we were to hold this survey many times, 90% of those intervals would contain the true population proportion, reflecting how individuals in the larger population decide not to go to college due to unaffordable costs.