Activity 19

MATH 216: Statistical Thinking

Confidence Intervals for Population Means

Time Allocation: 15 minutes total (5 min reading, 10 min individual work)

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions:

  1. What is the correct interpretation of a 95% confidence interval for a population mean, and why can’t we say “there’s a 95% probability that the true mean is in this interval”?
  1. How does sample size affect the margin of error and width of a confidence interval?
  1. What is the relationship between confidence level and critical z-values (90%, 95%, 99% confidence)?

Part 2: Confidence Interval Calculations (4 minutes)

Calculate confidence intervals using formula: \(\bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}\)

95% Confidence Intervals:

  1. Medical study: \(\bar{x} = 85\), \(\sigma = 12\), \(n = 64\) CI =

  2. Quality control: \(\bar{x} = 50\), \(\sigma = 8\), \(n = 36\) CI =

  3. Market research: \(\bar{x} = 25\), \(\sigma = 9.5\), \(n = 125\) CI =

90% and 99% Confidence Intervals:

  1. Same data as #2: \(\bar{x} = 50\), \(\sigma = 8\), \(n = 36\) 90% CI = 99% CI =

Show your work for one calculation:

Part 3: Interpretation and Application (3 minutes)

Interpret the confidence intervals:

  1. For the medical study (95% CI for blood pressure reduction), what does the interval tell us about the true treatment effect?
  1. Compare the widths of the 90%, 95%, and 99% confidence intervals. What trade-off exists between confidence level and precision?

Critical Thinking: If we want to reduce the margin of error by half while keeping the same confidence level, how much should we increase the sample size?