Question 1198357: Hi! You can email me this eurireunir14@gmail.com for the Excel file. I am trying to help my brother who is having a hard time solving this.
PHSTAT solutions are required for problems 3.
3.
a. Refer to the Excel file student-survey.xls. Construct a 99% confidence interval for the true mean distance in miles of classroom to current residence (variable DR) of a college student at the University of Florida. Interpret the result. Draw the “flowchart of clouds” as demonstrated in the lectures with all the elements of the point estimation of a mean.
b. Refer to the Excel file student-survey.xls. Tabulate the results of the variable AA in the survey using PHSTAT. Construct a 99% confidence interval for the true proportion of UF students who support affirmative action. Interpret the result. Draw the “flowchart of clouds” as demonstrated in the lectures with all the elements of the point estimation of a proportion.
c. How large of a sample would have to be taken so that the confidence interval estimate in part (3b) will be within 0.02 of the true proportion with 99% confidence?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **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.
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