Chapter 10 Review: Estimation of Confidence Intervals for Proportions and Means

Chapter 10 Review: Estimation of Confidence Intervals for Proportions and Means

Overview of Confidence Intervals

• Focus on constructs of confidence intervals for both proportions and means.
• Utilize algorithms and TI-84 calculators for calculations.

Question 1: Population Proportion Analysis

• Key Data:
    - Sample size (
  $n$
    ): 634
    - Accidents reported: 29
• Find Proportion (P-hat):
    - Formula:
  $\hat{P} = \frac{x}{n} = \frac{29}{634}$
    - Result:
  $\hat{P} = 0.0457$
• Construct 99% Confidence Interval:
    - Confidence Level: 99% (or 0.99 decimal)
    - Using TI-84 for calculation:
      - Command: 1-Prop Z-Int
      - Inputs:
        - x = 29
        - n = 634
        - C-level = 0.99
    - Result: Lower Confidence Limit: 0.024;
      Upper Confidence Limit: 0.067
• Key Insights:
    - Lower Confidence Level: Low bound of the confidence interval.
    - Upper Confidence Level: High bound of the confidence interval.

Question 2: Mean Commute Distance Analysis

• Key Data:
    - Sample size (
  $n$
    ): 40
• Given Data: Commute distances recorded.
• Construct 90% Confidence Interval for Mean:
    - Confidence Level: 0.90
    - Use TI-84: Command 1-Sample T-Interval
      - Data input in L3.
    - For Calculation:
      - Inputs: 40 data points
    - Outcome: Mean: 8.8; Confidence Limits: 7.624 to 9.976

Question 3: Sample Size Determination for Proportions

• Objective: Estimate the percentage of bullied middle school students.
• Parameters:
    - Required Confidence Level: 95% (0.95)
    - Margin of Error: 4% (0.04)
• Sample Size Calculation:     1. Estimate Formula:
  $n = \frac{(Z_{\alpha/2})^2 \cdot p(1-p)}{E^2}$
• Basic Calculation:
     - Using P-hat = 0.5 (as not provided)
     - Using Standard Z-value for 95% confidence: 1.96
• Results:
    - Minimum Sample Size (n) = 601 students

Question 4: Proportion of Americans Driving Under Influence

• Objective: Estimate proportion of Americans (16-20) driving under the influence.
• Requirements: 99% confidence
• Calculations: Adjust parameters and repeat previous formula adjustments using Z|alpha for new parameters to yield 1037 participants needed to survey.

Question 5: Estimating Mean Weekly Sales

• Objective: Estimate the mean of weekly sales for athletic footwear.
• Parameters: 90% confidence level
• Standard Deviation: 1200, Margin of error: 400
• Results: Necessary sample size (n) = 25 for precision within margin constraints.

Question 6: Proportion of Students with Extracurricular Activities

• Parameters:
    - Sample Size (n): 200
    - Positive Responses (x): 63
    - Confidence Level: 90%
• Use similar CI structure as earlier.
• Resulting Interval: 0.261 to 0.369
    - Margin of error = approx. 0.054

Question 7: Estimated Mean of Marshmallows in Cereal

• Data:
    - Sample Size: 100
    - Mean: 251, Standard Deviation: 17
• Confidence Level: 95%
• Outcome Results: CI = 247.627 to 254.373

Question 8: GPA Estimation

• Parameters:
    - Sample Size: 36, Mean: 2.81 (GPA), SD: 0.616
• Confidence Level: 99%
• Calculation yields confidence limits based on new data inputs resulting in approximations between defined ranges.

Question 9: Sample Size for Dental Rates with 95% Confidence

• Parameters: Sigma given as 4.8, E: 1.0:
• Process to determine sample size:
• Standard computation using Z-values leads to adjustments in critical values and resultant n = 88.514 which rounds to 89.

Question 10: Mortgage Interest Rate Survey

• Parameters:
    - Sample Size: 20
    - Mean: 6.93, SD: 0.42
• Confirm adjustments for CI with 99% possibly leading to approximate ranges.

Conclusion

• Various methodologies provided comprehensive steps to compute confidence intervals catering to population parameters of proportion and mean, ensuring extensive practice across demographic survey-like scenarios. Use care and repeat calculations to ensure clarity and accuracy in both manual and calculational methods.