Unit 3 · Topic 3.11 Beta

Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions

To judge a claim with an interval for p_1 − p_2 , look at where 0, or the claimed value, sits.

Practice 4: Interpret Results

Part 1 · Hook

Why this matters

A college says weekly nudge emails help students finish online courses, and its study gives a 95% interval for the difference in finishing rates from 0.03 to 0.11. Another study of snack sales at two stores gives an interval from −0.04-0.04 to 0.06. One interval supports a claim and the other does not. The difference is where 0 sits.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A 95% interval for pp is from 0.44 to 0.52. Is 0.5 a plausible value of pp?

  1. Yes, it is inside the interval
  2. No, it is above the center
  3. No, the interval is too wide
  4. Only at 99% confidence
Show the answer

Values inside the interval are plausible.

  • Correct: Yes, it is inside the interval:
  • No, it is above the center:
  • No, the interval is too wide:
  • Only at 99% confidence:

2. What does the interval (p^1−p^2)±z∗SE(\hat{p}_1 - \hat{p}_2) \pm z^* \text{SE} estimate?

  1. p1−p2p_1 - p_2
  2. p^1−p^2\hat{p}_1 - \hat{p}_2
  3. p1p_1
  4. the standard error
Show the answer

An interval estimates a parameter: the true difference.

  • Correct: p1−p2p_1 - p_2:
  • p^1−p^2\hat{p}_1 - \hat{p}_2:
  • p1p_1:
  • the standard error:

Part 4 · See it

See it first

Three 95% intervals for a difference in proportions against a dashed line at 0: study A from 0.03 to 0.11 lies entirely above 0, study B from -0.04 to 0.06 contains 0, and study C from -0.15 to -0.02 lies entirely below 0.
Three 95% intervals for p1−p2p_1 - p_2. Where 0 falls decides the claim each one supports. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A difference of 0 means p1=p2p_1 = p_2ask whether 0 is a plausible value of p1−p2p_1 - p_2, that is, whether it is inside the interval
  2. Every value in the interval is greater than 0the interval gives convincing evidence that p1>p2p_1 > p_2 (less than 0: that p1<p2p_1 < p_2)
  3. The interval contains 0equal proportions are plausible, so there is not convincing evidence of a difference, and the interval does not prove there is none
  4. A claim names a value such as "15 points more"check that value the same way: inside the interval means plausible, outside means convincing evidence against it

Part 6 · Key ideas

Key ideas

  • Entirely above 0: convincing evidence p1>p2p_1 > p_2. Entirely below 0: convincing evidence p1<p2p_1 < p_2. Contains 0: no convincing evidence of a difference.
  • Write the claim with the reason first: "Because every value in the interval is greater than 0, the interval gives convincing evidence that ..."
  • A 95% interval and a two-sided test at α=0.05\alpha = 0.05 nearly always agree about whether 0 is plausible.
  • Containing 0 never proves the proportions are equal; very large samples can make even tiny differences convincing.

Part 7 · Misconception

A common mistake

The wrong idea: If the interval contains 0, the study proved that the two proportions are the same.

What actually happens: Containing 0 only means a difference of 0 is plausible. Other values in the interval are plausible too, so the right wording is "not convincing evidence of a difference".

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Three studies, three 95% intervals

Each study below compares two groups with a 95% confidence interval for p1−p2p_1 - p_2 (group 1 minus group 2). The conditions for inference were met in every study. (Invented data.)
95% confidence intervals for p1−p2p_1 - p_2
StudyGroup 1Group 2Proportion of95% interval
Astudents sent weekly nudge emailsstudents sent no emailsstudents who finish the online coursefrom 0.03 to 0.11
Bshoppers at store 1shoppers at store 2shoppers who buy a snack at checkoutfrom −0.04-0.04 to 0.06
Cseedlings in new soilseedlings in old soilseedlings that survive a monthfrom −0.15-0.15 to −0.02-0.02

1. Which study gives convincing evidence that p1p_1 is greater than p2p_2?

  1. Studies A and B
  2. Study A
  3. Study B
  4. Study C
Show the answer

An interval entirely above 0 is convincing evidence that p1−p2>0p_1 - p_2 > 0.

  • Studies A and B: Study B's interval reaches below 0, so it does not give convincing evidence of a positive difference.
  • Correct: Study A: Right: every value from 0.03 to 0.11 is positive, so every plausible difference has group 1 higher.
  • Study B: Study B's interval contains 0, so equal proportions are plausible.
  • Study C: Study C's interval is entirely negative: it suggests group 2 is higher, the other direction.

2. What does Study B's interval let you conclude?

  1. Store 1 has the higher proportion, since most of the interval is positive
  2. There is not convincing evidence of a difference: 0 is a plausible value
  3. The two stores have exactly the same proportion of snack buyers
  4. The study made an error, since an interval should not cross 0
Show the answer

Containing 0 means equal proportions are plausible: no convincing evidence of a difference, and no proof of equality.

  • Store 1 has the higher proportion, since most of the interval is positive: More of the interval lies above 0, but 0 is still inside it, so a difference of 0 is plausible.
  • Correct: There is not convincing evidence of a difference: 0 is a plausible value: Right: the interval holds negative values, 0 and positive values, so it cannot rule out equal proportions.
  • The two stores have exactly the same proportion of snack buyers: An interval that contains 0 does not show the difference is 0; it shows 0 cannot be ruled out. Values up to 0.06 are plausible too.
  • The study made an error, since an interval should not cross 0: Intervals cross 0 often. It means the data are consistent with no difference.

3. Complete the claim about Study C in context.

Write the conclusion in context.

Show the answer

An interval entirely below 0 is convincing evidence that p1<p2p_1 < p_2.

  • A full-credit answer: Because every value in the interval from -0.15 to -0.02 is less than 0, the interval gives convincing evidence that the true proportion of seedlings that survive a month is lower in new soil than in old soil.

4. Study A's team also computes a 99% interval from the same data. What would it show?

  1. About 0.01 to 0.09, with both ends above 0
  2. About 0.017 to 0.123, with both ends above 0
  3. The same interval, from 0.03 to 0.11
  4. About −0.01-0.01 to 0.15, now containing 0
Show the answer

Same center, larger z∗z^*: the 99% interval is wider but here still excludes 0.

  • About 0.01 to 0.09, with both ends above 0: A higher confidence level makes the interval wider, not narrower.
  • Correct: About 0.017 to 0.123, with both ends above 0: Right: the 95% margin, 0.04, means SE=0.041.96=0.0204\text{SE} = \frac{0.04}{1.96} = 0.0204; the 99% margin is 2.576(0.0204)=0.0532.576(0.0204) = 0.053, and the lower end is still positive.
  • The same interval, from 0.03 to 0.11: The level changes z∗z^* from 1.96 to 2.576, so the interval widens.
  • About −0.01-0.01 to 0.15, now containing 0: The 99% margin is about 0.053, not 0.08. The lower end stays above 0.

5. A 95% interval for p1−p2p_1 - p_2 is from 0.01 to 0.08. What does this say about a two-sided test of H0:p1−p2=0H_0: p_1 - p_2 = 0 at α=0.05\alpha = 0.05 with the same data?

  1. The test would fail to reject H0H_0, because the interval is narrow
  2. The test would reject H0H_0 at α=0.01\alpha = 0.01 as well
  3. Nothing, because tests and intervals answer different questions
  4. The test would reject H0H_0, because 0 is not a plausible value
Show the answer

A C%C\% interval and a two-sided test at α=1−C\alpha = 1 - C give matching answers about a null value, except in rare borderline cases.

  • The test would fail to reject H0H_0, because the interval is narrow: Width is not the point. What matters is that 0 lies outside the interval.
  • The test would reject H0H_0 at α=0.01\alpha = 0.01 as well: A 99% interval would be needed for that, and it might reach 0. You cannot tell from the 95% interval.
  • Nothing, because tests and intervals answer different questions: They are closely linked: a 95% interval holds, nearly exactly, the values a two-sided test at 0.05 would not reject.
  • Correct: The test would reject H0H_0, because 0 is not a plausible value: Right: a two-sided test at α=0.05\alpha = 0.05 and a 95% interval nearly always agree; an interval well clear of 0 matches a p-value below 0.05.

6. A 95% interval for p1−p2p_1 - p_2 is from 0.07 to 0.19. Does it give convincing evidence for the claim that p1p_1 is more than 5 percentage points above p2p_2?

  1. No: you would need a test, not an interval
  2. Yes: every value in the interval is greater than 0.05
  3. No: the interval does not contain 0.05
  4. Yes: the sample difference, 0.13, is greater than 0.05
Show the answer

Compare the claimed value with the whole interval, not with the estimate.

  • No: you would need a test, not an interval: An interval can judge a claim about a value: check where the value sits relative to the interval.
  • Correct: Yes: every value in the interval is greater than 0.05: Right: the claim is p1−p2>0.05p_1 - p_2 > 0.05, and every plausible value is above 0.05.
  • No: the interval does not contain 0.05: Not containing 0.05 is exactly what supports the claim here, because the whole interval is above it.
  • Yes: the sample difference, 0.13, is greater than 0.05: The sample difference alone could be chance. The whole interval being above 0.05 is the evidence.

Part 9 · Summary

Summary

To judge a claim with an interval for p1−p2p_1 - p_2, look at where 0, or the claimed value, sits. Entirely above or below 0 gives convincing evidence of a difference in that direction; an interval that contains 0 gives no convincing evidence of a difference, which is not proof of equality. State the reason, the evidence and the parameter in context.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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