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CIE A-Level Maths Study Notes

5.4.6 Confidence Intervals for Population Mean

Understanding confidence intervals is integral to the field of statistics, particularly when it comes to estimating population means. These intervals are not just a measure of where a mean might lie but are a cornerstone of statistical inference, providing insights into the reliability of an estimate.

Confidence Intervals

Confidence intervals (CI) provide a range where we expect the true population parameter, like the mean, to be, given a certain confidence level. This is a way to measure how certain we are about an estimate.

Confidence interval illustration

Image courtesy of inchcalculator

Key Points:

  • Confidence Interval: A range expected to contain the true population parameter.
  • Components: Lower limit, upper limit, confidence level (e.g., 95%).
  • Use: Helps quantify uncertainty in estimates.

When Population Variance is Known:

  • Formula: CI=xˉ±z×σnCI = \bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}
  • Variables: xˉ\bar{x} = sample mean, zz= z-score for confidence level, σ\sigma = population standard deviation, nn = sample size.

Examples

Problem 1: Known Population Standard Deviation

Given:

  • Sample Mean (xˉ):68(\bar{x}): 68
  • Population Standard Deviation (σ):15(\sigma): 15
  • Sample Size (n):50(n): 50
  • Confidence Level: 95% (with zz ≈ 1.96)

Objective:

Find the 95% confidence interval for the average.

Solution:

1. Calculate Standard Error (SE):

  • Formula: SE=σnSE = \frac{\sigma}{\sqrt{n}}
  • Calculation: SE=15502.12SE = \frac{15}{\sqrt{50}} \approx 2.12

2. Determine Margin of Error (ME):

  • Formula: ME=z×SEME = z \times SE
  • Calculation: ME=1.96×2.124.16ME = 1.96 \times 2.12 \approx 4.16

3. Construct the Confidence Interval (CI):

  • Lower Limit: xˉME=684.16=63.84\bar{x} - ME = 68 - 4.16 = 63.84
  • Upper Limit: xˉ+ME=68+4.16=72.16\bar{x} + ME = 68 + 4.16 = 72.16

Thus, the 95% confidence interval for the population mean ranges from approximately 63.8463.84 to 72.1672.16.

Problem 2: Unknown Population Variance, Large Sample Size

Given Data:

  • Sample Mean (xˉ):170cm(\bar{x}): 170 cm
  • Sample Standard Deviation (s):20cm(s): 20 cm
  • Sample Size (n):100(n): 100
  • Confidence Level: 95% (Z-score ≈ 1.96)

Objective:

Determine the 95% confidence interval for the population mean height.

Solution:

1. Calculate the Standard Error (SE): SE=sn=20100=2 cmSE = \frac{s}{\sqrt{n}} = \frac{20}{\sqrt{100}} = 2 \text{ cm}

2. Calculate the Margin of Error (ME): ME=Z×SE=1.96×2=3.92 cmME = Z \times SE = 1.96 \times 2 = 3.92 \text{ cm}

3. Construct the Confidence Interval (CI): CI=xˉ±ME=170±3.92CI = \bar{x} \pm ME = 170 \pm 3.92

CI=[166.08 cm,173.92 cm]\therefore CI = [166.08 \text{ cm}, 173.92 \text{ cm}]

Thus, the 95% confident that the true population mean height is between 166.08 cm and 173.92 cm.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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