Question 1198357
**a) Confidence Interval for Mean Distance to Residence**

1. **Data Analysis:**
   - Use Excel to calculate the sample mean (x̄) and sample standard deviation (s) of the "DR" column (distance to residence).

2. **Confidence Interval:**
   - **Formula:** x̄ ± t*(s/√n) 
      * where:
          * x̄ is the sample mean
          * s is the sample standard deviation
          * n is the sample size
          * t is the critical t-value from the t-distribution table 

   - **Determine Degrees of Freedom (df):** df = n - 1 
   - **Find Critical t-value:** Use a t-distribution table or Excel function (e.g., T.INV.2T(0.01, df)) to find the critical t-value for a 99% confidence level and the appropriate degrees of freedom.

   - **Calculate the Margin of Error:** Margin of Error = t * (s/√n)

   - **Construct the Confidence Interval:** 
      * Lower Limit: x̄ - Margin of Error
      * Upper Limit: x̄ + Margin of Error

3. **Interpretation:**
   - The confidence interval provides a range of values within which we are 99% confident that the true mean distance to residence for all UF students lies.

**Flowchart of Clouds (Mean)**

* **Population:** All UF students
* **Parameter:** μ (Population Mean Distance to Residence)
* **Statistic:** x̄ (Sample Mean Distance to Residence)
* **Sampling Distribution:** t-distribution (approximately)
* **Confidence Level:** 99%
* **Critical Value:** tα/2 (from t-distribution table)
* **Standard Error:** s/√n
* **Confidence Interval:** x̄ ± tα/2 * (s/√n)

**b) Confidence Interval for Proportion of Students Supporting Affirmative Action**

1. **Data Analysis:**
   - Use Excel and PHSTAT to:
      - Tabulate the frequencies of "Yes" and "No" responses for the "AA" variable.
      - Calculate the sample proportion (p̂) of students who support affirmative action: p̂ = (Number of "Yes" responses) / (Sample Size)

2. **Confidence Interval:**
   - **Formula:** p̂ ± Z * √[p̂(1-p̂)/n] 
      * where:
          * p̂ is the sample proportion
          * Z is the critical z-value from the standard normal distribution (for 99% confidence, Z ≈ 2.576)
          * n is the sample size

   - **Calculate the Margin of Error:** Margin of Error = Z * √[p̂(1-p̂)/n]

   - **Construct the Confidence Interval:** 
      * Lower Limit: p̂ - Margin of Error
      * Upper Limit: p̂ + Margin of Error

3. **Interpretation:**
   - The confidence interval provides a range of values within which we are 99% confident that the true proportion of UF students who support affirmative action lies.

**Flowchart of Clouds (Proportion)**

* **Population:** All UF students
* **Parameter:** p (Population Proportion Supporting Affirmative Action)
* **Statistic:** p̂ (Sample Proportion Supporting Affirmative Action)
* **Sampling Distribution:** Approximately Normal (for large enough sample size)
* **Confidence Level:** 99%
* **Critical Value:** Zα/2 (from standard normal distribution)
* **Standard Error:** √[p̂(1-p̂)/n]
* **Confidence Interval:** p̂ ± Zα/2 * √[p̂(1-p̂)/n]

**c) Sample Size for Proportion with Margin of Error 0.02**

* **Formula:** n = (Z² * p̂ * (1-p̂)) / E² 
    * where:
        * n is the required sample size
        * Z is the critical z-value (2.576 for 99% confidence)
        * p̂ is the estimated sample proportion (use the value from part b)
        * E is the desired margin of error (0.02)

* **Calculate the required sample size** using the formula.

**Note:**

* This analysis requires access to the "student-survey.xls" file to perform the calculations and construct the confidence intervals.
* You can use Excel functions like AVERAGE, STDEV, and CONFIDENCE.NORM for the calculations.
* PHSTAT is an Excel add-in that can be used to perform statistical analyses, including hypothesis testing and confidence interval calculations.

I hope this helps! Let me know if you have any other questions.