Lesson 11: Estimation with Confidence Intervals

October 7, 2026

Review:

• Exam 2
• Sampling Distributions

Presentation:

• How to Estimate a Population Mean (μ)
  • Conduct a Simple Random Sample (SRS) of size n
  • Calculate the sample mean, x̅ = (∑x)/n
  • Calculate the sample standard deviation, s = √(∑(x – x̅)²/(n-1)
  • If n is sufficiently large (using n≥30 as our rule of thumb), we can
    • Assume the sampling distribution of x̅ is approximately Normal
    • Estimate the population mean, μ = x̅ (this is our point estimate)
    • Calculate margin of error, m = z*SE
      • where z = 1.96 for 95% confidence
      • where standard error = SE = s/(√n)
    • So, estimation of a population mean with confidence interval = x̅ ± z*SE
• Watch Against All Odds 
  • Video Unit 24, Confidence Intervals
• Calculation Demonstration
  • A sample of size n = 400 produced the sample mean, x̅ = 36.0 and standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
    • n = 400
    • x̅ = 36.0
    • s = 9.0
    • z = 1.96
    • SE = s/(√n) = 9/(√400)
    • m = z*SE = 1.96*(9/(√400)) = 0.882
    • Estimate of population mean = 36.0 ± 0.882
    • or 95% confidence interval: [35.118, 36.882]
    • Interpretation: We are 95% confident that the population mean is between 35.118 and 36.882.
• Click here for hand written examples

Activity:

• Complete the following Practice Problems
• Problem 1. A sample of size n = 100 produced the sample mean, x̅ = 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 sample of 100 customers at a grocery store had an average checkout time of 7.8 minutes with a standard deviation of 2.5 minutes. Construct a 95% confidence interval to estimate the population mean checkout time.
• Problem 4. A manufacturing supervisor records the time required for 144 workers to complete a task. The sample mean is 18.4 minutes and the sample standard deviation is 4.8 minutes. Construct a 95% confidence interval for the population mean completion time.
• Problem 5. A random sample of 64 college students reports an average of 6.7 hours of sleep per night, with a standard deviation of 1.6 hours. Construct a 95% confidence interval for the population mean number of hours of sleep.
• Problem 6. A sample of 225 households has an average monthly electricity bill of $138 with a standard deviation of $45. Construct a 95% confidence interval to estimate the population mean monthly electricity bill.
• Problem 7. A sample of 49 restaurant customers spent an average of $31.40 per meal with a standard deviation of $7.00. Construct a 95% confidence interval to estimate the mean amount spent.
• Problem 8. Two researchers study average commute time. Both obtain a sample mean of 28 minutes and a sample standard deviation of 12 minutes. Researcher A uses a sample of n=36 people. Researcher B uses a sample of n=144 people.
  • Before calculating, which researcher should have the smaller margin of error? Why?
  • Construct a 95% confidence interval for each researcher.