Comparing two groups: two proportions and paired data

About this lesson

Algebra II › Statistical inference

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Comparing two groups: two proportions and paired data

Warm-up

Three quick questions from earlier lessons.

Discover

To test whether two groups have the same success rate, group A had 30 of 100 and group B 50 of 100. Under H₀ (same rate), what is the best estimate of that common rate?

  1. 0.4
  2. It cannot be estimated

Key idea

Two independent groups, or one group measured twice.

Two proportions: pooled p^=x1+x2n1+n2\hat{p} = \dfrac{x_1 + x_2}{n_1 + n_2}, z=p^1−p^2p^(1−p^)(1/n1+1/n2)z = \dfrac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(1/n_1 + 1/n_2)}}. Paired data (same subjects twice): a one-sample t test on the differences.

Worked example

x1=30x_1 = 30, n1=100n_1 = 100; x2=50x_2 = 50, n2=100n_2 = 100.

p^=0.4\hat{p} = 0.4
Pool.
SE≈0.069SE \approx 0.069
√(0.4 × 0.6 × 0.02).
z≈−2.89z \approx -2.89
(0.3 − 0.5)/0.069.

Your turn, guided

Your turn: 20 of 50 and 30 of 50. The pooled proportion.

0.50.5
50/100.

Practice ★

  1. To test H₀: p₁ = p₂, group 1 had 21 successes out of 82 and group 2 had 22 out of 102. Find the pooled proportion.
  2. To test H₀: p₁ = p₂, group 1 had 51 successes out of 85 and group 2 had 51 out of 105. Find the pooled proportion.
  1. 24 students take a test before and after a course. Which procedure compares the scores?
    1. A paired t test on the 24 differences (after − before)
    2. A two-sample t test treating the groups as independent
    3. A two-proportion z test

Watch out!

Practice ★★

  1. Testing H₀: p₁ = p₂ with n₁ = n₂ = 100, the pooled proportion is 3/10. Find the standard error of p̂₁ − p̂₂.

Exit ticket

Three questions to check you have got it.

  1. To test H₀: p₁ = p₂, group 1 had 21 successes out of 82 and group 2 had 22 out of 102. Find the pooled proportion.
  2. To test H₀: p₁ = p₂, group 1 had 51 successes out of 85 and group 2 had 51 out of 105. Find the pooled proportion.
  3. To test H₀: p₁ = p₂, group 1 had 32 successes out of 93 and group 2 had 53 out of 113. Find the pooled proportion.

Summary card

Comparing two groups

  • Two proportions: pooled p^\hat{p} in the SE.
  • Paired: test the differences.

Watch out: pool counts, not proportions.

End of the lesson

Next lesson: Observational studies, experiments and confounding Back to the chapter

Answers: Comparing two groups: two proportions and paired data

A

  1. 43184\frac{43}{184}
  2. 5195\frac{51}{95}

B

  1. A paired t test on the 24 differences (after − before)

C

  1. 42100\frac{\sqrt{42}}{100}

D

  1. 43184\frac{43}{184}
  2. 5195\frac{51}{95}
  3. 85206\frac{85}{206}

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