What if a two-proportion confidence interval has 0?

Q: Does an interval containing 0 mean the two proportions are equal?

No. It means 0 is one plausible value among all the values between the endpoints, so singling it out is not justified. Values near the center of the interval fit your data better than values near the endpoints, but every value in the range is one you cannot rule out. The correct conclusion is that you do not have convincing evidence of a difference.

Q: My 95% interval contains 0. Would a 99% interval also contain it?

Yes. A higher confidence level uses a larger critical value, so the interval is wider around the same center and still swallows 0. The reverse is not guaranteed: a 90% interval is narrower and could exclude 0. A conclusion that flips when you change the level is a borderline one, so report the level you actually used.

Q: Can I read a p-value off the confidence interval?

No. An interval gives a decision at one significance level for a two-sided alternative, not a p-value. If the question asks for a p-value, for a one-sided conclusion, or for a formal test, run the two-proportion z test and compare its p-value to your significance level.

Q: Why did my interval and my z test give different answers?

The interval uses each sample's own proportion in the standard error, while the two-proportion z test pools the samples because the null hypothesis assumes the proportions are equal. The two standard errors are slightly different, so results sitting right on the boundary can disagree. Report whichever procedure the question asked for.

By Jude Wallis · Published July 28, 2026

It means 0 is a plausible value for the difference, so you do not have convincing evidence that the two proportions differ. It does not mean they are equal. The interval also supports every other value between its endpoints, and you cannot rule those out either.

AP Statistics: Unit 3 (topics 3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions, 3.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions). This is Unit 3 topics 3.10 and 3.11 of the Fall 2026 AP Statistics course, where you build an interval for the difference between two population proportions and justify a claim from it.

What 0 inside the interval means

A confidence interval for $p_1 - p_2$ is a range of plausible values for the true difference between two population proportions. If 0 sits inside that range, then "the two proportions are equal" is one possibility your data cannot rule out.

That is the whole answer, and it is smaller than students expect. The interval does not say the difference is 0. It says 0 survived, along with every other number between the endpoints.

Take the interval $(-0.024, 0.114)$ for $p_1 - p_2$ . Plausible values include $-0.02$ (group 2 ahead by about 2 percentage points), $0$ (a dead tie), and $0.11$ (group 1 ahead by about 11 percentage points). All three are consistent with what you observed. You have not learned that the groups match. You have learned that your data are not precise enough to say which group is ahead, or by how much.

Why "there is no difference" is the wrong conclusion

Here is the sentence that loses points: "Since the interval contains 0, there is no difference between the two proportions."

It fails for a simple reason. Your interval holds many values and 0 is only one of them. Pulling 0 out of the range and announcing it as the truth ignores every other value the same data support. If you are willing to claim the difference is 0 because 0 is inside, you should be nearly as willing to claim it is 0.11, because 0.11 is inside too. Values near the middle of the interval do fit the data best, but that gives 0 no special claim on being the truth.

The honest move is to describe what you failed to establish. Failing to rule out 0 is not the same as ruling in 0. Statisticians put it as absence of evidence versus evidence of absence: not finding a difference is not the same as finding no difference.

The same trap shows up in testing, where you never accept $H_0$ , you only fail to reject it. An interval that contains 0 is that fail-to-reject verdict in a different form. If a real difference does exist and your sample missed it, you have committed a Type II error, which is covered in Type I vs Type II errors.

Read both endpoints, not just the zero

The signs of the two endpoints carry the whole story, and 0 is the dividing line.

Both endpoints positive: 0 is not plausible and every plausible difference is positive, so you have convincing evidence that $p_1 > p_2$ .
Both endpoints negative: every plausible difference is negative, so you have convincing evidence that $p_1 < p_2$ .
One endpoint negative and one positive: 0 is inside, so the data are consistent with group 1 ahead, group 2 ahead, or a tie. You cannot even name a direction.

That last line is the part students skip. When the interval straddles 0, you have not shown the groups are similar. You have shown nothing about direction at all.

Two mechanical warnings. First, the order of subtraction is yours to pick, but state it and keep it. If you define the difference as $p_1 - p_2$ and get $(-0.024, 0.114)$ , then defining it the other way around gives $(-0.114, 0.024)$ : the endpoints swap and change sign. Whether 0 is inside never changes, but the direction of your conclusion does. Second, compare the endpoints as signed numbers. The interval $(-0.30, -0.05)$ does not contain 0, even though $-0.05$ is close to it.

How this lines up with a two-sided test

There is a direct correspondence and the AP course expects you to use it. A confidence interval for $p_1 - p_2$ that contains 0 goes with failing to reject $H_0: p_1 = p_2$ in a two-sided test at the matching significance level. Written as a decimal, the matching level is $\alpha = 1 - C$ , where $C$ is the confidence level.

Confidence level	Matching two-sided level	0 inside the interval means
90%	$\alpha = 0.10$	fail to reject $H_0$ at $\alpha = 0.10$
95%	$\alpha = 0.05$	fail to reject $H_0$ at $\alpha = 0.05$
99%	$\alpha = 0.01$	fail to reject $H_0$ at $\alpha = 0.01$

Three cautions come with that table.

The match holds for a two-sided alternative only. A 95% interval says nothing directly about a one-sided test at $\alpha = 0.05$ , so if $H_a$ points one way, run the test. See one-tailed vs two-tailed tests.

For proportions the match is close but not exact. The interval builds its standard error from each sample's own proportion, while the two-proportion z test pools the samples because $H_0$ says the proportions are equal. Those two standard errors differ slightly, so in a borderline case the interval and the test can disagree. When a question asks for a test, report the test. The difference in setup is laid out in one-proportion vs two-proportion z test.

An interval gives a decision, never a p-value. It tells you the verdict at one level. If the question wants a p-value, compute one with the proportion z test calculator or by hand, and compare it to $\alpha$ as described in p-value vs alpha.

What to write on the exam

Topic 3.11 asks you to justify a claim from an interval for $p_1 - p_2$ , and graders look for a specific shape. Fill in the blanks:

"Because the ___% confidence interval from ___ to ___ contains 0, it is plausible that there is no difference between [parameter 1 in context] and [parameter 2 in context]. We do not have convincing evidence that the two differ."

Filled in for the worked example below: "Because the 95% confidence interval from $-0.024$ to $0.114$ contains 0, it is plausible that there is no difference between the true proportion of District A seniors with a part-time job and the true proportion of District B seniors with one. We do not have convincing evidence that the two districts differ."

The pieces that earn credit:

Name the confidence level and both endpoints.
Say that 0 is a plausible value for the difference.
Use "not convincing evidence" rather than "no difference" or "the same".
Name both populations and the variable in context, not just $p_1$ and $p_2$ .

The phrasings that cost you: "there is no difference", "the proportions are equal", "we accept $H_0$ ", and "there is a 95% chance the difference is 0". That last one misreads the confidence level, which describes the method rather than one finished interval, as explained in what 95% confidence means. For more on rubric wording, see the AP Statistics FRQ guide.

Containing 0 is a statement about precision

Whether 0 falls inside depends on two things: how far apart your sample proportions landed, and how wide the margin of error is.

ME = z^*\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}

The sample sizes sit in the denominators under the square root, so the margin of error shrinks roughly in proportion to $1/\sqrt{n}$ . Quadruple both sample sizes and the margin roughly halves. Keep the same sample proportions and gather more data, and an interval that contained 0 can stop containing it. Worked example 2 does exactly that: identical proportions, three times the data, and 0 drops out.

A higher confidence level pushes the other way. A 99% interval from the same data is wider than the 95% one, so if the 95% interval contains 0, the 99% interval does too.

So "contains 0" is not a fact about the two populations. It is a fact about your evidence. Two studies of the same populations can disagree about whether 0 is inside, and the study with more data is the one able to detect a small real difference. The full sequence of two-proportion inference lives in Unit 3.

A 95% interval for the difference that contains 0

A random sample of 400 seniors from District A finds 200 who hold a part-time job. An independent random sample of 400 seniors from District B finds 182 who hold one. Construct a 95% confidence interval for $p_1 - p_2$ , where $p_1$ is the true proportion for District A, and state what it supports.

Define the parameter. Let $p_1$ be the true proportion of all District A seniors with a part-time job and $p_2$ the same proportion for District B. You are estimating $p_1 - p_2$ .
Check conditions. Random: both samples were selected at random and independently. 10%: the problem does not give district sizes, so state the assumption. Each district must have at least 4,000 seniors for a sample of 400 to be less than 10% of its population. Large counts: the observed successes and failures are 200 and 200 for District A, and 182 and 218 for District B, all at least 10.
Sample proportions. $\hat{p}_1 = \frac{200}{400} = 0.5$ and $\hat{p}_2 = \frac{182}{400} = 0.455$ .
Point estimate. $\hat{p}_1 - \hat{p}_2 = 0.5 - 0.455 = 0.045$ .
Standard error, without pooling (an interval uses each sample's own proportion). First piece: $\frac{(0.5)(0.5)}{400} = \frac{0.25}{400} = 0.000625$ . Second piece: $(0.455)(0.545) = 0.247975$ , and $\frac{0.247975}{400} = 0.00061994$ . Sum: $0.000625 + 0.00061994 = 0.00124494$ . Square root: $\sqrt{0.00124494} = 0.035284$ .
Critical value. For 95% confidence with the standard Normal, $z^* = 1.96$ .
Margin of error. $1.96 \times 0.035284 = 0.069156$ , about 0.0692.
Interval. $0.045 \pm 0.0692$ , giving $(-0.0242, 0.1142)$ .
Read the endpoints. One is negative and one is positive, so 0 is inside. Plausible values run from District B ahead by about 2 percentage points to District A ahead by about 11 percentage points.
Cross-check with the matching test. Pooling gives $\hat{p}_c = \frac{200 + 182}{800} = 0.4775$ , so $SE = \sqrt{(0.4775)(0.5225)\left(\frac{1}{400} + \frac{1}{400}\right)} = \sqrt{(0.24949)(0.005)} = 0.035320$ and $z = \frac{0.045}{0.035320} = 1.27$ . The two-sided p-value is about $0.20$ , far above 0.05, so you fail to reject $H_0: p_1 = p_2$ . Same verdict as the interval.

The 95% confidence interval for $p_1 - p_2$ is about $(-0.024, 0.114)$ . Because it contains 0, it is plausible that the two districts have the same true proportion of seniors with a part-time job, so there is not convincing evidence of a difference. Concluding that the proportions are equal would be wrong: the same interval supports a District A lead as large as 0.114.

Same proportions, three times the data, and 0 drops out

Repeat the comparison with samples three times as large and identical sample proportions: 600 of 1200 District A seniors and 546 of 1200 District B seniors hold a part-time job. Build the 95% confidence interval for $p_1 - p_2$ and compare it to worked example 1.

Check conditions again. The samples are still random and independent, and the counts 600 and 600 for District A and 546 and 654 for District B are all at least 10. The 10% condition is now a stronger assumption: each district needs at least 12,000 seniors for a sample of 1200 to stay under 10% of its population.
Sample proportions are unchanged. $\hat{p}_1 = \frac{600}{1200} = 0.5$ and $\hat{p}_2 = \frac{546}{1200} = 0.455$ , so the point estimate is still $0.5 - 0.455 = 0.045$ .
Standard error. First piece: $\frac{0.25}{1200} = 0.00020833$ . Second piece: $\frac{0.247975}{1200} = 0.00020665$ . Sum: $0.00041498$ . Square root: $\sqrt{0.00041498} = 0.020371$ .
Margin of error. $1.96 \times 0.020371 = 0.039927$ , about 0.0399.
Interval. $0.045 \pm 0.0399$ , giving $(0.0051, 0.0849)$ . Both endpoints are positive, so 0 is not inside.
Compare with worked example 1. The center did not move at all. Tripling both sample sizes divided the standard error by $\sqrt{3} \approx 1.732$ , shrinking the margin of error from 0.0692 to 0.0399, which is exactly what pushed 0 out.
The matching test agrees. The pooled proportion is still $\frac{600 + 546}{2400} = 0.4775$ , the pooled standard error is $\sqrt{(0.24949)\left(\frac{1}{1200} + \frac{1}{1200}\right)} = 0.020392$ , so $z = \frac{0.045}{0.020392} = 2.21$ and the two-sided p-value is about 0.027. Since $0.027 < 0.05$ , reject $H_0$ .

The 95% interval is about $(0.005, 0.085)$ , which does not contain 0. With the same sample proportions but three times the data, there is now convincing evidence that District A has the higher proportion. Whether 0 lands inside your interval depends on your sample sizes, not only on the populations.

Frequently asked questions

Does an interval containing 0 mean the two proportions are equal?

No. It means 0 is one plausible value among all the values between the endpoints, so singling it out is not justified. Values near the center of the interval fit your data better than values near the endpoints, but every value in the range is one you cannot rule out. The correct conclusion is that you do not have convincing evidence of a difference.

My 95% interval contains 0. Would a 99% interval also contain it?

Yes. A higher confidence level uses a larger critical value, so the interval is wider around the same center and still swallows 0. The reverse is not guaranteed: a 90% interval is narrower and could exclude 0. A conclusion that flips when you change the level is a borderline one, so report the level you actually used.

Can I read a p-value off the confidence interval?

No. An interval gives a decision at one significance level for a two-sided alternative, not a p-value. If the question asks for a p-value, for a one-sided conclusion, or for a formal test, run the two-proportion z test and compare its p-value to your significance level.

Why did my interval and my z test give different answers?

The interval uses each sample's own proportion in the standard error, while the two-proportion z test pools the samples because the null hypothesis assumes the proportions are equal. The two standard errors are slightly different, so results sitting right on the boundary can disagree. Report whichever procedure the question asked for.