To determine the required sample size for estimating the proportion of freshmen who do not visit their counselors regularly, we can use the formula for the sample size in estimating proportions:

$n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right)$

Where:

• $n$ = required sample size
• $Z$ = Z-value corresponding to the desired confidence level
• $p$ = estimated proportion of the population
• $E$ = margin of error

Step 1: Identify the values

• Desired confidence level: 98%
• $p = 0.31$
• $E = 0.015$ (1.5%)

Step 2: Find the Z-value

For a 98% confidence level, the Z-value can be found from the standard normal distribution. The critical value $Z$ for 98% confidence is approximately:

$Z \approx 2.33$

Step 3: Plug values into the formula

$n = \left( \frac{(2.33)^2 \cdot (0.31) \cdot (1 - 0.31)}{(0.015)^2} \right)$

Step 4: Calculate each component

1. Calculate $Z^2$:

$Z^2 = (2.33)^2 \approx 5.4289$

2. Calculate $p \cdot (1 - p)$:

$p \cdot (1 - p) = 0.31 \cdot (1 - 0.31) = 0.31 \cdot 0.69 \approx 0.2149$

3. Calculate $E^2$:

$E^2 = (0.015)^2 = 0.000225$

Step 5: Substitute back into the formula

$n = \frac{5.4289 \cdot 0.2149}{0.000225}$

$n = \frac{1.16530}{0.000225} \approx 5175.56$

Step 6: Round up to the nearest whole number

Since we cannot have a fraction of a sample, we round up:

$n \approx 5176$

Conclusion

The required sample size to estimate the proportion of freshmen who do not visit their counselors regularly, with 98% confidence and within 1.5%, is 5176.

If you need any further details or have questions, feel free to ask! Here are some related questions to consider:

1. How would the sample size change if the desired confidence level was reduced to 95%?
2. What if the margin of error was increased to 3%?
3. How would you conduct the sampling to ensure it is representative?
4. What are the potential limitations of this sample size calculation?
5. How can you verify if the actual proportion of freshmen aligns with your sample estimate?

Tip: Always ensure that your sample is random to avoid bias in your estimates.