Confidence Interval Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
A sample of n=64n=64 has xห‰=85\bar{x}=85 and s=16s=16. Construct a 95% confidence interval for the population mean.

Solution

  1. 1
    Standard error: SE=sn=1664=168=2SE = \frac{s}{\sqrt{n}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2
  2. 2
    Critical value for 95% CI: zโˆ—=1.96z^* = 1.96
  3. 3
    Margin of error: E=zโˆ—ร—SE=1.96ร—2=3.92E = z^* \times SE = 1.96 \times 2 = 3.92
  4. 4
    95% CI: xห‰ยฑE=85ยฑ3.92=(81.08,88.92)\bar{x} \pm E = 85 \pm 3.92 = (81.08, 88.92)

Answer

95% CI: (81.08,88.92)(81.08, 88.92). We are 95% confident the population mean is in this interval.
A 95% confidence interval means: if we repeated this sampling procedure many times, 95% of the resulting intervals would contain the true population mean. The interval does NOT say there's a 95% chance the mean is in this specific interval โ€” the mean is fixed, the interval is random.

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 2 hard

Compare 90% and 99% confidence intervals for [formula], [formula], [formula]. Calculate both and exp

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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