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Large-Sample Confidence Intervals worksheets

Use conservative proportion bounds, the approximate 95% rule, and CLT intervals for general means.

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State independence and large-sample conditions. Label conservative and rule-of-thumb intervals as approximations.

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Keep assumptions beside the interval

Every task states the sampling model, standard-error source, quantile convention, and rounding reference needed to reproduce the worked endpoint.

Learning objective

What this activity builds

Construct and compare large-sample intervals while stating independence, finite-variance, and approximation conditions.

Worked example

What is the conservative proportion margin at 95%?

1.96/(2 square root n), approximately 1/square root n.

The Bernoulli variance is at most one fourth.

Teaching notes

Make the practice count

  • Keep endpoints within zero and one.
  • Call the 1/square root n rule approximate.
  • Do not use n=30 as a universal CLT guarantee.

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Math SheetNameScoreTeacherDateLarge-Sample Confidence IntervalsState independence and large-sample conditions. Label conservative and rule-of-thumb intervals as approximations.1.A random independent sample of n=64 observations from a skewed population has x-bar=20 and s=4. Assuming finite variance and that n is large enough for theCLT approximation, construct a 90% mean interval.Reference: Approximate large-sample interval: x-bar +/- z_(1-alpha/2)s/sqrt(n), with z_(1-alpha/2)=1.6448536. Independence, finite variance, and adequate CLT accuracy are assumptions.2.31 successes occur in n=68 independent Bernoulli trials. Construct the conservative 95% normal interval using p(1-p)<=1/4.Reference: Use p-hat +/- z_(1-alpha/2)/(2sqrt(n)), with z_(1-alpha/2)=1.9599639; clip endpoints to [0,1]. This bound is conservative for the normal-approximation standard error.State assumptions and the quantile convention; round only final endpoints.math-sheet.comPage 1 of 4
Math SheetNameScoreTeacherDateLarge-Sample Confidence IntervalsState independence and large-sample conditions. Label conservative and rule-of-thumb intervals as approximations.3.Two comparable independent proportion studies use n=72 and n=288. Under the same conservative 95% approximation, compare margin widths and state whennominal coverage may be unreliable.Reference: Conservative margin is about 1/sqrt(n), so multiplying sample size by 4 halves the margin. Normal coverage can be poor with dependence or inadequate success/failure counts.4.36 of n=76 independent trials succeed. Use the approximate 95% rule p-hat +/- 1/sqrt(n), and label the result as a rule of thumb.Reference: The 1/sqrt(n) margin approximates the conservative 95% normal margin because 1.96/2 is about 1. It is not exact.State assumptions and the quantile convention; round only final endpoints.math-sheet.comPage 2 of 4
Math SheetNameScoreTeacherDateLarge-Sample Confidence IntervalsState independence and large-sample conditions. Label conservative and rule-of-thumb intervals as approximations.5.39 successes occur in n=80 independent Bernoulli trials. Construct the conservative 95% normal interval using p(1-p)<=1/4.Reference: Use p-hat +/- z_(1-alpha/2)/(2sqrt(n)), with z_(1-alpha/2)=1.9599639; clip endpoints to [0,1]. This bound is conservative for the normal-approximation standard error.6.42 of n=84 independent trials succeed. Use the approximate 95% rule p-hat +/- 1/sqrt(n), and label the result as a rule of thumb.Reference: The 1/sqrt(n) margin approximates the conservative 95% normal margin because 1.96/2 is about 1. It is not exact.State assumptions and the quantile convention; round only final endpoints.math-sheet.comPage 3 of 4
Math SheetNameScoreTeacherDateLarge-Sample Confidence IntervalsState independence and large-sample conditions. Label conservative and rule-of-thumb intervals as approximations.7.A random independent sample of n=88 observations from a skewed population has x-bar=26 and s=10. Assuming finite variance and that n is large enough for theCLT approximation, construct a 90% mean interval.Reference: Approximate large-sample interval: x-bar +/- z_(1-alpha/2)s/sqrt(n), with z_(1-alpha/2)=1.6448536. Independence, finite variance, and adequate CLT accuracy are assumptions.8.Two comparable independent proportion studies use n=92 and n=368. Under the same conservative 95% approximation, compare margin widths and state whennominal coverage may be unreliable.Reference: Conservative margin is about 1/sqrt(n), so multiplying sample size by 4 halves the margin. Normal coverage can be poor with dependence or inadequate success/failure counts.State assumptions and the quantile convention; round only final endpoints.math-sheet.comPage 4 of 4

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