Lesson 9: Estimation of Population Parameters with Confidence Intervals

September 24, 2024

Review:

• Sampling Distributions
• Grades posted on Bb
  • Exam 1 (200 points possible)
  • Attendance/Participation, “AP 1” (30 points possible)
• NFL Game Scoring Data

Presentation:

• How to Estimate a Population Mean (μ)
  • Conduct a Simple Random Sample (SRS) of size n
  • Calculate the sample mean, x-bar = (∑x)/n
  • Calculate the sample standard deviation, s = √(∑(x – x-bar)²/(n-1)
  • If n is sufficiently large (≥30), we can
    • Assume x-bar is Normally distributed
    • Estimate the population mean, μ = x-bar (point estimate)
    • Calculate margin of error, m = z*σ
      • where z = 1.96 for 95% confidence
      • where σ = s/(√n)
  • Estimation with confidence interval for μ = x-bar ± m
• Watch Against All Odds Video Unit 24, Confidence Intervals

• Calculation Demonstration
  • A sample of size n = 400 produced the sample mean, x-bar = 36.0 and standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
    • n = 400
    • x-bar = 36.0
    • s = 9.0
    • m = 1.96*(9/(√400)) = 0.882
    • Estimate for μ = 36.0 ± 0.882
    • 95% confidence interval: [35.118, 36.882]

Activity:

• Problem 1. A sample of size n = 100 produced the sample mean, x-bar = 16.0 and standard deviation, s = 3.0. Construct a 95% confidence interval to estimate the population mean.
• Problem 2. An operation manager at a large plant observed 120 workers assembling an electronic component. The average time needed for assembly was 16.2 minutes with a standard deviation of 3.6 minutes. Construct a 95% confidence interval to estimate the mean assembly time.
• Problem 3. A computer technician installs new hard drives on 64 different computers. The average installation time is 42 minutes with a standard deviation of 5 minutes. Construct a 95% confidence interval for the mean installation time.
• Problem 4. A research firm conducted a survey of regular smokers to estimate the average amount spent per week on cigarettes. A sample of 49 regular smokers revealed average spending on cigarettes to be $21.55 with a standard deviation of $5.21. Construct a 95% confidence interval to estimate mean weekly cigarette spending.
• Problem 5. This week on Thursday Night Football the New York Giants will host the Dallas Cowboys. The “over-under” is set at 44.5 points. According to pro-football-reference.com, in all NFL games (n=1,086 games) the total score is normally distributed with a mean total score of 52.9 and a standard deviation of 20.2. Download the data here. What’s the probability that the total score will be less than 44.5? Will this information help with the over-under bet? What other data would be helpful?

Assignment:

• Repeat Problems 1-5 (above) in Google Sheets or Microsoft Excel
• Add to your portfolio
• Read pp. 353-362, Introduction to Inference and Estimation with Confidence