Sketching the P-Value Region for a Test

Let the Alternative Choose the Shading

A p-value sketch makes the logic of a hypothesis test visible: it shows which outcomes would count as at least as extreme as the observed result if the null hypothesis were true. The key decision is which outcomes count as extreme. As in Choosing One-Sided or Two-Sided Alternatives, the alternative hypothesis—not the direction of the sample result—determines which tail or tails to shade.

For a one-proportion \(z\)-test, the null model for \(\hat{p}\) is approximately Normal when the test conditions are met. It is centered at the null proportion \(p_0\), with standard error \(SE_0=\sqrt{p_0(1-p_0)/n}\). The standardized scale uses \(z=0\) for that same center. The observed sample proportion \(\hat{p}\) and its test statistic \(z_{\mathrm{obs}}\) mark corresponding locations on these two scales.

Definition: The p-value region is the part of the null model containing results at least as extreme as the observed result in the direction or directions specified by \(H_a\). Its area is the p-value.

First calculate \(\hat{p}=x/n\), then calculate \(z_{\mathrm{obs}}=(\hat{p}-p_0)/SE_0\), as in The Test Statistic z for One Proportion. A positive \(z_{\mathrm{obs}}\) means \(\hat{p}\) is above \(p_0\); a negative \(z_{\mathrm{obs}}\) means it is below \(p_0\). But the sign does not select the tail. The alternative does.

How to Mark and Shade the Null Model

Sketch a Normal curve, identify its center, and mark the observed statistic on the horizontal axis. On the standard Normal scale, the center is \(z=0\). On the sample-proportion scale, the center is \(\hat{p}=p_0\). If using the proportion scale, also mark the observed \(\hat{p}\). Then shade the region specified by \(H_a\).

Sketching rule: For \(H_a:p<p_0\), shade at or below \(z_{\mathrm{obs}}\). For \(H_a:p>p_0\), shade at or above \(z_{\mathrm{obs}}\). For \(H_a:p\ne p_0\), shade both tails beyond \(z_{\mathrm{obs}}\) and its equally distant mirror point across 0.

Alternative	Region to shade on the z scale	What to label
\(H_a:p<p_0\)	All \(z\) values at or below \(z_{\mathrm{obs}}\)	\(0\), \(z_{\mathrm{obs}}\), and the lower-tail area
\(H_a:p>p_0\)	All \(z\) values at or above \(z_{\mathrm{obs}}\)	\(0\), \(z_{\mathrm{obs}}\), and the upper-tail area
\(H_a:p\ne p_0\)	Both tails at least \(\lvert z_{\mathrm{obs}}\rvert\) from 0	\(0\), \(z_{\mathrm{obs}}\), the mirrored cutoff, and both tail areas

A useful check is to imagine moving from the observed statistic outward in the direction named by the alternative. For a left-tailed test, “outward” means farther left; for a right-tailed test, it means farther right. For a two-sided test, move away from 0 on both sides. This still works when the observed statistic is opposite the alternative: a right-tailed test with a negative \(z_{\mathrm{obs}}\) shades everything to the right of that negative value, which can be most of the curve.

On a sample-proportion scale, the same locations can be shown using the null standard error. The observed proportion is \(p_0+z_{\mathrm{obs}}SE_0\). For a two-sided test, the mirrored cutoff is \(p_0-\lvert z_{\mathrm{obs}}\rvert SE_0\) if the observed statistic is positive, and \(p_0+\lvert z_{\mathrm{obs}}\rvert SE_0\) if it is negative. These cutoffs show which sample proportions correspond to the shaded tails.

\hat{p}_{\mathrm{obs}}=p_0+z_{\mathrm{obs}}SE_0

The p-value region is not the same as a rejection region. A p-value sketch shades results at least as extreme as the particular observed statistic. A rejection region is set by a chosen significance level \(\alpha\) and a critical value. Do not shade to a critical value when the task is to sketch the p-value.

Worked Examples

Worked Example: A Left-Tailed Test with a Negative Statistic

A fictional transit agency randomly selects 400 riders from a list of 10,000 riders who use a particular route. In the sample, 180 report using a mobile ticket. Test whether the true proportion of riders on this list who use a mobile ticket is less than 0.50. Use \(\alpha=0.05\), and describe the p-value sketch.

State: Let \(p\) be the true proportion of riders on this list who use a mobile ticket. The hypotheses are \(H_0:p=0.50\) and \(H_a:p<0.50\).

Plan and check conditions: Use a one-proportion \(z\)-test. The riders were randomly selected, so the Random condition is met. The sample was drawn without replacement from 10,000 riders, and \(0.10(10{,}000)=1{,}000\); since \(400\leq1{,}000\), the 10% condition is met. Under the null, the expected number of mobile-ticket users is \(400(0.50)=200\), and the expected number of other riders is \(400(0.50)=200\). Both are at least 10, so the Large Counts condition is met.

Do: Calculate the observed sample proportion, null standard error, and test statistic:

\hat{p}=\frac{180}{400}=0.45, \qquad SE_0=\sqrt{\frac{0.50(0.50)}{400}} =\sqrt{0.000625} =0.025

z_{\mathrm{obs}}=\frac{0.45-0.50}{0.025}=-2.00

On the standard Normal sketch, mark the center at 0 and the observed statistic at \(-2.00\). Since the alternative is left-tailed, shade the area at or to the left of \(-2.00\). On the proportion scale, mark the null center at 0.50 and the observed proportion at 0.45, then shade at or below 0.45. The shaded area is:

\text{p-value}=P(Z\leq-2.00) =\operatorname{normalcdf}(-1\text{E}99,-2.00,0,1) \approx0.0228

Conclude: Assuming the true proportion is 0.50, the probability of getting a test statistic at or below \(-2.00\) is approximately 0.0228. Since \(0.0228<0.05\), reject \(H_0\). The sample provides convincing evidence that the true proportion of riders on this list who use a mobile ticket is less than 0.50.

Worked Example: A Right-Tailed Test with a Negative Statistic

A fictional parks department randomly selects 300 households from a list of 5,000 households near a proposed trail. Of those selected, 135 say they would use the trail. The question is whether the true proportion of households on the list that would use the trail is greater than 0.50. Describe where to shade the p-value region.

The sample proportion is \(\hat{p}=135/300=0.45\). Under \(H_0:p=0.50\), the null standard error is:

SE_0=\sqrt{\frac{0.50(0.50)}{300}} =\sqrt{0.0008333333} \approx0.0288675

The hypotheses are \(H_0:p=0.50\) and \(H_a:p>0.50\). The sample is random. Since \(0.10(5{,}000)=500\) and \(300\leq500\), the 10% condition is met. The null expected counts are \(300(0.50)=150\) households who would use the trail and \(300(0.50)=150\) who would not; both are at least 10. Thus the one-proportion \(z\)-test conditions are met.

The observed statistic is:

z_{\mathrm{obs}}=\frac{0.45-0.50}{0.0288675}\approx-1.7321

Here the data point opposite the direction in \(H_a\): the observed sample proportion is below 0.50 even though the alternative asks whether \(p\) is greater. Still, shade to the right of the observed statistic, because \(H_a:p>0.50\). The cutoff is \(-1.7321\), not \(+1.7321\). On the proportion scale, shade at or above 0.45. The area is large:

\text{p-value}=P(Z\geq-1.7321) =\operatorname{normalcdf}(-1.7321,1\text{E}99,0,1) \approx0.9584

This example shows why the sample’s direction cannot determine the tail. Shading left of \(-1.7321\) would answer a left-tailed question, not the stated right-tailed one.

Worked Example: A Two-Tailed Test and Two Shaded Regions

A fictional adult-learning center randomly selects 400 enrolled learners from a list of 5,000. In the sample, 220 say they prefer evening classes. Test whether the true proportion of learners on the list who prefer evening classes differs from 0.50. Use \(\alpha=0.05\), and describe the sketch.

Let \(p\) be the true proportion of learners on this list who prefer evening classes. The hypotheses are \(H_0:p=0.50\) and \(H_a:p\ne0.50\). The sample is random; \(400\leq0.10(5{,}000)=500\); and the null expected counts are \(400(0.50)=200\) successes and \(400(0.50)=200\) failures. Thus the Random, 10%, and Large Counts conditions are met.

The sample proportion and null standard error are:

\hat{p}=\frac{220}{400}=0.55, \qquad SE_0=\sqrt{\frac{0.50(0.50)}{400}}=0.025

The test statistic is \(z_{\mathrm{obs}}=(0.55-0.50)/0.025=2.00\). Mark 0 at the center, \(2.00\) to its right, and \(-2.00\) as the equally distant point on the left. Because \(H_a\) is two-sided, shade both areas: \(Z\leq-2.00\) and \(Z\geq2.00\). On the proportion scale, the null center is 0.50, the observed proportion is 0.55, and the matching two cutoffs are 0.45 and 0.55.

\text{p-value} =P(Z\leq-2.00)+P(Z\geq2.00) \approx0.02275013+0.02275013 \approx0.04550026 \approx0.0455

The tail areas are approximately 0.0227501 each; adding before rounding gives 0.0455003, which rounds to 0.0455. Since \(0.0455<0.05\), reject \(H_0\). The sample provides convincing evidence that the true proportion of learners on this list who prefer evening classes differs from 0.50.

Common Mistakes and AP Exam Tips

A sketch can be neat and still be wrong if the shaded tail does not match the alternative. Before drawing, write down whether \(H_a\) says less than, greater than, or different from \(p_0\). Then mark the observed statistic and shade according to that direction.

Choosing a tail from the sign of \(z_{\mathrm{obs}}\). The sign tells you whether \(\hat{p}\) is above or below \(p_0\); it does not choose the p-value tail. In a right-tailed test, shade right of \(z_{\mathrm{obs}}\) even if that statistic is negative.
Shading the wrong side of the cutoff. For \(H_a:p<p_0\), shade left of the observed statistic. For \(H_a:p>p_0\), shade right of it. The cutoff itself is the observed statistic, not zero.
Shading only the observed side in a two-sided test. When \(H_a:p\ne p_0\), show both tails beyond the observed distance from zero, including the mirrored cutoff.
Confusing the observed result with a critical value. The p-value sketch uses \(z_{\mathrm{obs}}\). A significance-level rejection region uses a cutoff determined by \(\alpha\); these answer different questions.
Leaving the sketch unlabeled. Label the null center, \(z_{\mathrm{obs}}\), and the shaded region. If using the proportion scale, label \(p_0\) and \(\hat{p}\) as well. This makes the direction and meaning of the shaded area clear.

AP Exam Tip: A strong sketch identifies the null model, its center, the observed statistic, and every tail included by \(H_a\). In a written response, state that the shaded area is calculated assuming \(H_0\) is true. For a one-sided test, make clear that the tail follows the alternative even when the observed statistic has the opposite sign.

Key Takeaway

The sketch is a picture of the p-value under the null model. Mark the observed statistic, then let the alternative—not the observed sign—tell you which side or sides to shade. When helpful, translate the same landmarks to the sample-proportion scale using \(p_0\) as the null center.

Key takeaway: Left-tailed means shade at or below \(z_{\mathrm{obs}}\); right-tailed means shade at or above it; two-sided means shade both tails at least \(\lvert z_{\mathrm{obs}}\rvert\) from zero.

Check Your Understanding

For each question, focus on the null-model sketch and the region counted as the p-value.

For \(H_0:p=0.40\) and \(H_a:p<0.40\), where should the p-value region be shaded relative to \(z_{\mathrm{obs}}\)?
A right-tailed test has \(z_{\mathrm{obs}}=-1.10\). Which side of \(-1.10\) should be shaded, and why?
A two-sided test has \(z_{\mathrm{obs}}=1.60\). What are the two z-scale cutoffs for the shaded tails?
On a sample-proportion sketch, what value is at the center of the null model: \(\hat{p}\) or \(p_0\)?
Explain one difference between the p-value region and a rejection region defined by \(\alpha\).