What is the standard error?

Answer: 0.0295 SE = sqrt(0.32·0.68/250) = sqrt(0.000871) ≈ 0.0295.

What is the 99% CI for p?

Answer: (0.112, 0.188) SE = sqrt(0.15·0.85/600) ≈ 0.01458. ME = 2.576·0.01458 ≈ 0.0376. CI: 0.15 ± 0.0376 = (0.112, 0.188).

Using z = 1.282, what is the 80% CI?

Answer: (0.241, 0.359) SE = sqrt(0.3·0.7/100) ≈ 0.0458. ME = 1.282·0.0458 ≈ 0.0588. CI ≈ (0.241, 0.359).

AP Statistics · Topic 6.2

Constructing a Confidence Interval for a Population Proportion Practice

Part of Inference for Categorical Data: Proportions.(UNC-4.A)

Practice questions

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Sample questions

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Sample 1difficulty 2/5
$A sample of 250 has p̂ = 0.32.$
$What is the standard error?$
- A
  $0.0148$
- B
  $0.0009$
- C
  $0.0295$
  check_circle
- D
  $0.0590$
Why
$SE = sqrt(0.32\cdot0.68/250) = sqrt(0.000871) \approx 0.0295.$
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Sample 2difficulty 2/5
$A sample of 600 patients shows 90 had a side effect. Construct a 99% CI for the population proportion.$
$What is the 99% CI for p?$
- A
  $(0.090, 0.210)$
- B
  $(0.140, 0.160)$
- C
  $(0.122, 0.178)$
- D
  $(0.112, 0.188)$
  check_circle
Why
$SE = sqrt(0.15\cdot0.85/600) \approx 0.01458. ME = 2.576\cdot0.01458 \approx 0.0376. CI: 0.15 \pm 0.0376 = (0.112, 0.188).$
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Sample 3difficulty 2/5
$Sample of n = 100, p̂ = 0.30. Construct an 80% CI.$
$Using z* = 1.282, what is the 80% CI?$
- A
  $(0.250, 0.350)$
- B
  $(0.241, 0.359)$
  check_circle
- C
  $(0.260, 0.340)$
- D
  $(0.215, 0.385)$
Why
$SE = sqrt(0.3\cdot0.7/100) \approx 0.0458. ME = 1.282\cdot0.0458 \approx 0.0588. CI \approx (0.241, 0.359).$
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Sample 4difficulty 2/5
$A random sample of 400 voters finds 240 support a ballot measure. Conditions for inference are met.$
$Which is the 95% confidence interval for the population proportion?$
- A
  $(0.560, 0.640)$
- B
  $(0.500, 0.700)$
- C
  $(0.540, 0.660)$
- D
  $(0.552, 0.648)$
  check_circle
Why
$p̂ = 0.60. SE = sqrt(0.6\cdot0.4/400) = 0.0245. ME = 1.96\cdot0.0245 \approx 0.048. CI: 0.60 \pm 0.048 = (0.552, 0.648).$
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Sample 5difficulty 2/5
$In a sample of n=500, p̂=0.40. We want a 95% CI for p.$
$What is the margin of error?$
- A
  $0.086$
- B
  $0.196$
- C
  $0.022$
- D
  $0.043$
  check_circle
Why
$SE = sqrt(0.4\cdot0.6/500) \approx 0.0219. ME = 1.96\cdot0.0219 \approx 0.043.$
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