What It Means When the Interval Contains Zero

When Zero Is a Plausible Difference

In Interpreting a Confidence Interval for \(p_1-p_2\), you learned to read the endpoints as plausible values for a difference between two population proportions. Now focus on a special value: zero. If the difference is zero, the two population proportions are equal. So when zero lies inside a confidence interval, equality remains a plausible value based on the interval.

That does not show that the proportions are equal. The interval may also contain negative and positive values, representing plausible differences in either direction. The appropriate conclusion is that the interval does not establish a convincing difference between the population proportions at the confidence level used.

Key idea: For an interval estimating \(p_1-p_2\), zero represents no difference between the population proportions. If the interval contains zero, the interval includes no difference among its plausible values, so it does not provide convincing evidence of a difference in either direction at the corresponding level of confidence.

The sample proportions can still differ. For example, \(\hat{p}_1-\hat{p}_2\) might be positive even when the interval extends below zero. The sample difference is the point estimate; the interval accounts for sampling variability. If zero is among its plausible values, the observed difference is not precise enough, relative to that variability and confidence level, to rule out no population difference.

Connecting the Interval to a Two-Sided Test

The corresponding two-sided test asks whether the population proportions differ in either direction. Using the same parameter definition and group order as the interval, its hypotheses are:

H_0:p_1-p_2=0 \qquad\text{versus}\qquad H_a:p_1-p_2\ne0

The null hypothesis says there is no difference; the alternative says there is a difference, without specifying which group has the higher proportion. In this test, zero is the null value. A two-sided test’s p-value measures how unusual the observed difference, or a difference at least as extreme in either direction, would be if the population proportions were equal.

A confidence interval and a two-sided test answer closely related questions. A confidence interval asks which parameter values are plausible given the data and the interval method. A test asks whether the data provide convincing evidence against a particular null value. When zero is inside an interval, the interval does not rule out the null value of zero. Consistent with that, a corresponding two-sided test generally does not reject the null at the matching significance level.

Connection: For procedures that are matched to one another, a confidence interval containing the null value corresponds to failing to reject the two-sided test at the significance level associated with that confidence level. For instance, a 95% interval is associated with a two-sided test at \(\alpha=0.05\).

There is a technical detail for two proportions. The usual two-proportion confidence interval uses an unpooled standard error, calculated from the two sample proportions separately. The usual two-proportion \(z\)-test uses a pooled standard error under the null hypothesis that the proportions are equal. Because these standard errors differ, the interval and test are not exact mathematical inverses in every borderline case. In most examples their conclusions agree, but do not claim that their numerical decisions must always match exactly. If a question asks for a formal test, carry out the test with its own conditions, statistic, and p-value.

Worked Examples

Worked Example: A Two-Sided Test and a 95% Interval

Setting: Imagine independent random samples of customers from two regions. A success is a customer who uses a mobile payment app at least once a week. In Region 1, 52 of 100 sampled customers report weekly use; in Region 2, 45 of 100 do. Assume each region has at least 1,000 customers. Consider a 95% confidence interval for \(p_1-p_2\) and a two-sided test at \(\alpha=0.05\).

State: Let \(p_1\) and \(p_2\) be the true proportions of customers in Regions 1 and 2, respectively, who use a mobile payment app at least once a week. The test is \(H_0:p_1-p_2=0\) versus \(H_a:p_1-p_2\ne0\).

Plan: The two samples are stated to be independent random samples, supporting the Random condition and independence between groups. Each sample of 100 is at most 10% of its population because each region has at least 1,000 customers. For the interval’s Large Counts condition, Region 1 has 52 successes and \(100-52=48\) failures; Region 2 has 45 successes and \(100-45=55\) failures. All four counts are at least 10.

For the two-sided test, the pooled proportion is \(\hat{p}_{\text{pool}}=(52+45)/(100+100)=0.485\). The expected successes and failures in each group under the null are \(100(0.485)=48.5\) and \(100(0.515)=51.5\), all at least 10. The test’s Large Counts condition is also met.

Do: The sample difference is:

\hat{p}_1-\hat{p}_2 =\frac{52}{100}-\frac{45}{100} =0.52-0.45 =0.07

For the 95% interval, use the unpooled standard error:

\begin{aligned} SE_{\hat{p}_1-\hat{p}_2} &=\sqrt{\frac{0.52(0.48)}{100}+\frac{0.45(0.55)}{100}}\\ &=\sqrt{0.002496+0.002475}\\ &=\sqrt{0.004971}\approx0.07051 \end{aligned}

With \(z^*\approx1.96\), the margin of error is \(1.959964(0.0705053)\approx0.13819\). Thus:

0.07\mathbin{\pm}0.13819 \quad\Longrightarrow\quad (-0.06819,\ 0.20819)

Zero lies inside this interval. For the two-sided test, use the pooled standard error:

\begin{aligned} SE_0 &=\sqrt{0.485(0.515)\left(\frac{1}{100}+\frac{1}{100}\right)} \approx0.07068\\ z &=\frac{0.52-0.45}{0.07068}\approx0.9904 \end{aligned}

The two-sided p-value is approximately \(0.3220\), rounded. Since \(0.3220>0.05\), we fail to reject \(H_0\).

Conclude: The interval includes zero, so no difference is among the plausible values for the true difference in weekly app use between the two regions. The two-sided test also fails to reject equality at the 0.05 significance level. There is not convincing evidence that the true proportions differ. This does not prove they are equal; the interval also includes positive and negative values.

Worked Example: A Small Sample Difference with a Wide Interval

Setting: Imagine independent random samples of students from two schools. A success is a student who participates in an after-school arts activity. At School A, 18 of 60 sampled students participate; at School B, 15 of 60 do. Assume each school has at least 600 students. Find and interpret a 95% confidence interval for \(p_1-p_2\), with School A first.

State and plan: Let \(p_1\) and \(p_2\) be the true proportions of students at Schools A and B, respectively, who participate in an after-school arts activity. The interval estimates \(p_1-p_2\). The samples are stated to be independent random samples, supporting randomness and independence between groups. Each sample is at most 10% of its school’s population. The success and failure counts are 18 and 42 at School A, and 15 and 45 at School B. All four counts are at least 10, so the interval’s Large Counts condition is met.

Do: The sample proportions differ by \(0.05\):

\hat{p}_1-\hat{p}_2 =\frac{18}{60}-\frac{15}{60} =0.30-0.25 =0.05

Calculate the unpooled standard error and the 95% interval:

\begin{aligned} SE_{\hat{p}_1-\hat{p}_2} &=\sqrt{\frac{0.30(0.70)}{60}+\frac{0.25(0.75)}{60}}\\ &=\sqrt{0.003500+0.003125}\\ &=\sqrt{0.006625}\approx0.08139\\ 0.05\mathbin{\pm}1.959964(0.0813941) &\approx0.05\mathbin{\pm}0.15953\\ &\Longrightarrow(-0.10953,\ 0.20953) \end{aligned}

Conclude: We are 95% confident that the true proportion of students participating in an after-school arts activity at School A minus the true proportion at School B is between about \(-0.110\) and \(0.210\). Zero lies inside the interval. Differences in either direction, as well as no difference, are plausible values according to this interval. The sample proportion at School A is higher, but this interval does not give convincing evidence that the population proportion at School A is higher or lower.

Worked Example: Zero Inside a 99% Interval

Setting: Imagine independent random samples of households from two towns. A success is a household that composts food scraps at home. In Town 1, 64 of 120 sampled households compost; in Town 2, 58 of 120 do. Assume each town has at least 1,200 households. Find and interpret a 99% confidence interval for \(p_1-p_2\), with Town 1 first.

State and plan: Let \(p_1\) and \(p_2\) be the true proportions of households in Towns 1 and 2, respectively, that compost food scraps at home. The samples are independent random samples, and each sample is at most 10% of its town’s households. The success and failure counts are 64 and 56 in Town 1, and 58 and 62 in Town 2. Each count is at least 10, so the Large Counts condition is satisfied.

Do: The point estimate is \(64/120-58/120=0.05\). The standard error is:

\begin{aligned} SE_{\hat{p}_1-\hat{p}_2} &=\sqrt{\frac{(64/120)(56/120)}{120} +\frac{(58/120)(62/120)}{120}}\\ &=\sqrt{0.00207407+0.00208102}\\ &=\sqrt{0.00415509}\approx0.06446 \end{aligned}

For 99% confidence, \(z^*\approx2.575829\). The margin of error is \(2.575829(0.06446)\approx0.16604\), giving:

0.05\mathbin{\pm}0.16604 \quad\Longrightarrow\quad (-0.11604,\ 0.21604)

Conclude: We are 99% confident that the true proportion of households that compost in Town 1 minus the true proportion in Town 2 is between about \(-0.116\) and \(0.216\). Since zero is inside the interval, this 99% interval does not establish a convincing difference. The interval is wider than a lower-confidence interval would be because the higher confidence level requires a larger margin of error; it therefore includes a broad range of possible differences.

Common Mistakes and AP Exam Tips

Claiming the proportions are equal: Zero being inside the interval means equality is plausible, not proven. A full-credit conclusion says there is not convincing evidence of a difference, rather than saying the groups are the same.
Ignoring the interval’s other values: An interval such as \((-0.07,0.21)\) includes negative values as well as zero and positive values. The data do not establish which population proportion is larger.
Confusing the point estimate with the conclusion: A positive \(\hat{p}_1-\hat{p}_2\) does not by itself establish that \(p_1>p_2\). Discuss the interval’s uncertainty, not just the observed sample difference.
Calling the result a test rejection: If the task asks only for an interval interpretation, answer in terms of plausible values and whether zero is included. If it asks for a formal test, state the hypotheses, check the test conditions, calculate the test statistic and p-value, and make a decision.
Assuming every interval and test calculation is identical: The usual two-proportion interval and test use different standard errors. Their conclusions commonly agree, but borderline results can differ. Use the requested procedure and avoid asserting exact equivalence unless the methods are matched.
Reversing the subtraction order: For \(p_1-p_2\), subtract Group 2's proportion from Group 1's: Group 1 minus Group 2. Preserve that order when interpreting negative or positive endpoints.

AP Exam Tip: Identify zero as the no-difference value for \(p_1-p_2\). If it lies inside the interval, explain that no difference remains plausible and that the interval does not provide convincing evidence of a difference at the corresponding level. Do not say the null is proved or accepted.

Key Takeaway

Zero in an interval for \(p_1-p_2\) means that no difference between the population proportions is one of the plausible values. The interval therefore does not establish a convincing difference in either direction. This interpretation connects to a two-sided test of \(H_0:p_1-p_2=0\), but it is not proof that the proportions are equal.

Key takeaway: If zero is inside the interval, report that no difference remains plausible and avoid claiming equality. For a formal two-sided test, use the test procedure and its own conditions and calculations.

Check Your Understanding

Use the meaning of zero and the group order to answer each question.

An interval for \(p_1-p_2\) is \((-0.04,0.15)\). What does the inclusion of zero tell you about evidence for a difference?
Why does an interval containing zero not prove that the two population proportions are equal?
Write the null and alternative hypotheses for a two-sided test of whether two population proportions differ.
A 95% interval contains zero. What significance level is associated with the corresponding two-sided test when the procedures are matched?
Why might a usual two-proportion interval and a usual two-proportion test give different decisions in a borderline case?