SH1, SH3 - Confidence intervals with known variance

SH1, SH3 - Confidence intervals with known variance

Construct symmetric confidence intervals for a normal population mean when its variance is known. You will connect the interval to the sampling distribution, interpret confidence correctly and use a constructed or given interval to assess a proposed mean.

Constructing a normal-mean interval

A sample mean is 102, but how wide a range should we report for the population mean? An estimate without its uncertainty can look more precise than the data justify.

Suppose 25 independent measurements are randomly sampled from a normal population whose standard deviation is known to be 10. The sample mean has standard deviation 10/25=210/\sqrt{25}=2. This is the standard error of the mean, describing how much sample means vary, rather than how much individual observations vary.

For a standard normal variable, the central 95% lies between approximately −1.960-1.960 and 1.9601.960. Applying these rounded boundaries to the mean and rearranging gives

P(−1.960≤Xˉ−μσ/n≤1.960)≈0.95P(Xˉ−1.960σn≤μ≤Xˉ+1.960σn)≈0.95\begin{aligned} \mathrm{P}\left(-1.960\le\frac{\bar X-\mu}{\sigma/\sqrt n}\le1.960\right)&\approx0.95\\ \mathrm{P}\left(\bar X-1.960\frac{\sigma}{\sqrt n}\le\mu\le\bar X+1.960\frac{\sigma}{\sqrt n}\right)&\approx0.95 \end{aligned}

The random quantity in this second line is the pair of endpoints, both based on the random sample.

Known-variance confidence interval

For n≥1n\ge1 independent random observations from N(μ,σ2)\mathrm{N}(\mu,\sigma^2), with known σ>0\sigma>0, a symmetric 100(1−α)%100(1-\alpha)\% confidence interval for μ\mu is

xˉ±z1−α/2σn\bar x\pm z_{1-\alpha/2}\frac{\sigma}{\sqrt n}

where Φ(z1−α/2)=1−α/2\Phi(z_{1-\alpha/2})=1-\alpha/2. The margin of error (half-width) is z1−α/2σ/nz_{1-\alpha/2}\sigma/\sqrt n.

The standardised normal sampling pivot is in the formula booklet; this rearranged interval formula is not printed separately. Familiar normal critical values appear in the infinite-degrees-of-freedom row of its t table; other quantiles can be found with an inverse-normal calculator function.

Normality makes this sampling result exact for any positive sample size; a large-sample threshold is not required when the population really is normal and σ\sigma is known. Unknown σ\sigma requires a different treatment taught next.

Worked example 1: a 95% interval

Question. For n=25n=25, xˉ=102\bar x=102 and known σ=10\sigma=10, construct a 95% confidence interval for the normal population mean.

Plan. Use the normal standard error and put 2.5% outside each endpoint.

SE⁡=10/5=2,102±1.960(2)=102±3.92\operatorname{SE}=10/5=2,\qquad102\pm1.960(2)=102\pm3.92

Answer. The 95% interval is (98.08,105.92)(98.08,105.92).

Check. Its midpoint is 102. Each endpoint is 3.92 from the midpoint, corresponding to 3.92/2=1.9603.92/2=1.960 standard errors.

Keep the sample and known standard deviation fixed; change only the confidence level to 99%. Greater coverage requires a wider interval.

Worked example 2: a higher confidence level

Question. Construct the 99% confidence interval from the same data, using z0.995=2.576z_{0.995}=2.576.

Plan. Use the same standard error and allow 0.5% in each tail.

102±2.576(2)=102±5.152102\pm2.576(2)=102\pm5.152

Answer. The interval is (96.848,107.152)(96.848,107.152), or (96.85,107.15)(96.85,107.15) to two decimal places.

Check. Its width is 10.30410.304, exceeding the earlier width 7.847.84. Their ratio 2.576/1.9602.576/1.960 equals the ratio of the critical values, as only confidence changed.

Watch out

Wrong: use 0.95 as the cumulative probability for a symmetric 95% interval. Correct: use (1+0.95)/2=0.975(1+0.95)/2=0.975. Also distinguish known variance 100 from its standard deviation 10.

Confidence describes the procedure

Imagine repeatedly taking independent samples of the same size from the same population, and constructing a 95% interval each time. Sample means change, so interval locations change. With known σ\sigma and fixed nn, their widths stay the same.

95% confidence

A 95% confidence procedure produces intervals that contain the fixed population mean in 95% of repeated samples under the stated assumptions. The percentage describes the procedure's long-run coverage.

[DIAGRAM: asset_name: Repeated confidence intervals; asset_slug: aqa7367-confidence-coverage; caption: Intervals move with the sample mean while the population mean stays fixed; description: Twenty illustrative 95 percent intervals with known standard error one are plotted horizontally against sample number. All have half-width 1.96. A vertical dashed line marks the fixed population mean 50. Nineteen intervals cross the line and one does not. The missed interval is clay and covering intervals are blue. A note says a particular batch need not contain exactly 95 percent successful intervals.]
Intervals move with the sample mean while the population mean stays fixed

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