Confidence Interval Math Example 2

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Example 2

hard
Compare 90% and 99% confidence intervals for xห‰=100\bar{x}=100, s=15s=15, n=36n=36. Calculate both and explain the trade-off between confidence and precision.

Solution

  1. 1
    SE =1536=2.5= \frac{15}{\sqrt{36}} = 2.5
  2. 2
    90% CI: zโˆ—=1.645z^* = 1.645; E=1.645ร—2.5=4.11E = 1.645 \times 2.5 = 4.11; CI: (95.89,104.11)(95.89, 104.11)
  3. 3
    99% CI: zโˆ—=2.576z^* = 2.576; E=2.576ร—2.5=6.44E = 2.576 \times 2.5 = 6.44; CI: (93.56,106.44)(93.56, 106.44)
  4. 4
    Trade-off: 99% CI is wider (12.88 vs 8.22 wide) โ€” more confidence requires sacrificing precision

Answer

90% CI: (95.89,104.11)(95.89, 104.11); 99% CI: (93.56,106.44)(93.56, 106.44). Higher confidence โ†’ wider, less precise interval.
There is an inherent trade-off: higher confidence requires a wider net. To be more confident we catch the true mean, we must cast a wider interval. The only way to get both high confidence AND high precision is to increase sample size.

About Confidence Interval

A range of values, computed from sample data, that is likely to contain the true population parameter with a specified level of confidence.

Learn more about Confidence Interval โ†’

More Confidence Interval Examples

Example 1 medium

A sample of [formula] has [formula] and [formula]. Construct a 95% confidence interval for the popul

Example 3 easy

A 95% CI for a population mean is [formula]. Find the sample mean [formula] and the margin of error

Example 4 hard

To achieve a margin of error of [formula] with 95% confidence, given [formula], find the required sa

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