Chi-Square Goodness-of-Fit Test: Legacy AP Guide
The chi square goodness of fit test compares one categorical variable’s observed counts with a fully specified distribution of expected proportions. College Board removed this procedure from the revised AP Statistics course beginning in 2026–27, so this page is retained for older released material, enrichment, and college-statistics continuity.
2026–27 course-status notice
Legacy topic: College Board lists chi-square goodness of fit among the topics removed from the revised AP Statistics course effective for the 2026–27 school year. Keep this page for historical released questions and enrichment, but do not treat it as current-core May 2027 exam content.
Chi-Square Goodness-of-Fit Test: Legacy AP Guide: direct answer
Directly stated for this lesson: The chi square goodness of fit test compares one categorical variable’s observed counts with a fully specified distribution of expected proportions. College Board removed this procedure from the revised AP Statistics course beginning in 2026–27, so this page is retained for older released material, enrichment, and college-statistics continuity.
The single primary keyword for this page is chi square goodness of fit test. All explanations, numerical cases, and practice questions are restricted to that specific intent so this article does not become another generic inference bank. Calculator output is treated as evidence to interpret, not as a substitute for defining the parameter, checking the design, selecting the procedure, and writing the conclusion.
Quick reference for chi square goodness of fit test
| Element | What to know |
|---|---|
| Course status | Removed from the revised AP Statistics course effective 2026–27; retained here as legacy/enrichment |
| Data | One categorical variable with counts in k categories |
| Null model | The population category proportions equal a specified distribution |
| Expected count | Eᵢ = n pᵢ |
| Statistic | χ² = Σ (O−E)²/E |
| Degrees of freedom | Usually k−1 when the category probabilities are fully specified |
| Condition | Expected counts should ordinarily be at least 5 for the chi-square approximation |
Concept mastery: chi square goodness of fit test
Recognize that this is now legacy AP content
College Board removed the chi-square goodness of fit test from the revised AP Statistics course effective for 2026–27. The procedure remains valuable for older released questions, general introductory statistics, and understanding how observed categorical counts can be compared with a specified model. Current AP preparation should prioritize the revised framework rather than treating this page as a current-core exam requirement.
Use one categorical variable and a specified distribution
Goodness of fit starts with counts for one categorical variable and null proportions supplied by theory, policy, historical expectation, or another fully specified model. If the question compares categorical distributions across multiple groups, homogeneity is usually the better framing. If it studies association between two variables in one population, independence is the better framing.
Convert null proportions into expected counts
For category i, Eᵢ=n pᵢ. Expected counts come from the null model, not from averaging observed counts unless equal proportions are actually part of H₀. The expected counts should sum to the total sample size; checking that sum is an efficient arithmetic safeguard.
Measure discrepancy cell by cell
Each category contributes (O−E)²/E to χ². Squaring removes sign, so the total statistic is nonnegative. Dividing by E scales the squared discrepancy relative to the count expected under H₀. A category with a large raw difference does not necessarily dominate if its expected count is also very large.
Use degrees of freedom from the number of categories
With k categories and fully specified null probabilities, the usual degrees of freedom are k−1 because once k−1 counts are known, the final count is determined by the fixed total. More advanced settings that estimate parameters from the same data can reduce df further, but older AP-style problems typically supplied the distribution directly.
Check expected counts, not observed counts, for the approximation
The standard chi-square approximation ordinarily requires expected counts to be at least 5. The condition is about E values generated by the null model. A small observed count is not automatically a violation if the expected count is adequate, though an extreme discrepancy may then contribute strongly to χ².
Use the upper tail only
Large χ² values indicate greater discrepancy between observed and expected counts, so the p-value is the upper-tail probability P(Χ²df≥observed χ²). There is no meaningful “negative chi-square” direction and no left-tailed goodness-of-fit alternative in the usual formulation.
Interpret rejection as lack of fit, not as a directional result
Rejecting H₀ indicates that the observed categorical distribution is inconsistent with the specified proportions. It does not by itself identify which category caused the difference. Examine standardized or component contributions to describe where the mismatch is strongest.
Respect sampling design and independence
Random sampling or another defensible mechanism is needed for population inference, and observations should be independent. Repeated counts from the same unit or clustered data may violate the simple multinomial model. The chi-square formula cannot correct a poor data-collection design.
Use component contributions diagnostically
After a significant result, calculate or inspect (O−E)²/E for each category. Large contributions identify cells that drive the total discrepancy. This is an explanatory follow-up, not a collection of separate hypothesis tests, and the signs of O−E are useful when describing whether observed counts were above or below expectation.
Distinguish legacy practice from current exam priority
Older AP Statistics free-response questions and textbooks may still contain goodness-of-fit tests. Students using historical materials should label the topic as legacy so time is not diverted from current assessable content. Teachers may still use it as an enrichment bridge to categorical inference or college course expectations.
Use technology after the expected-count table is understood
A calculator can evaluate χ² and the upper-tail p-value quickly, but the expected counts and df should be independently checkable. If lists are used, verify that observed and expected entries align category by category and that expected counts total n. A swapped category order can silently produce a meaningless statistic.
30 worked cases for chi square goodness of fit test
Legacy goodness-of-fit case 1: Food Cooperative Membership Drive
An older categorical-data exercise specifies 4 null category proportions 0.100, 0.200, 0.300, 0.400 for n=180 observations. The observed counts are [22, 31, 60, 67]. Under H₀, expected counts are n×pᵢ, giving approximately [18.0, 36.0, 54.0, 72.0]. These expected values sum to 180.0, matching the total sample size, and the smallest expected count is 18.0; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 2.597 with df=3, producing an upper-tail p-value of 0.4580. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 1: Food Cooperative Membership Drive.
Legacy goodness-of-fit case 2: Rural Broadband Project
An older categorical-data exercise specifies 5 null category proportions 0.073, 0.137, 0.200, 0.263, 0.327 for n=192 observations. The observed counts are [19, 20, 45, 43, 65]. Under H₀, expected counts are n×pᵢ, giving approximately [14.0, 26.2, 38.4, 50.6, 62.8]. These expected values sum to 192.0, matching the total sample size, and the smallest expected count is 14.0; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 5.593 with df=4, producing an upper-tail p-value of 0.2317. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 2: Rural Broadband Project.
Legacy goodness-of-fit case 3: City Tree Survival Audit
An older categorical-data exercise specifies 6 null category proportions 0.057, 0.101, 0.145, 0.189, 0.232, 0.276 for n=204 observations. The observed counts are [18, 14, 38, 29, 57, 48]. Under H₀, expected counts are n×pᵢ, giving approximately [11.6, 20.6, 29.5, 38.5, 47.4, 56.4]. These expected values sum to 204.0, matching the total sample size, and the smallest expected count is 11.6; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 13.532 with df=5, producing an upper-tail p-value of 0.0189. The data show substantial lack of fit to the specified distribution. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 3: City Tree Survival Audit.
Legacy goodness-of-fit case 4: Workforce Credential Program
An older categorical-data exercise specifies 4 null category proportions 0.123, 0.208, 0.292, 0.377 for n=216 observations. The observed counts are [34, 37, 72, 73]. Under H₀, expected counts are n×pᵢ, giving approximately [26.5, 44.8, 63.2, 81.5]. These expected values sum to 216.0, matching the total sample size, and the smallest expected count is 26.5; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 5.586 with df=3, producing an upper-tail p-value of 0.1336. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 4: Workforce Credential Program.
Legacy goodness-of-fit case 5: Regional Pharmacy Network
An older categorical-data exercise specifies 5 null category proportions 0.067, 0.133, 0.200, 0.267, 0.333 for n=228 observations. The observed counts are [23, 21, 56, 50, 78]. Under H₀, expected counts are n×pᵢ, giving approximately [15.2, 30.4, 45.6, 60.8, 76.0]. These expected values sum to 228.0, matching the total sample size, and the smallest expected count is 15.2; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 11.252 with df=4, producing an upper-tail p-value of 0.0239. The data show substantial lack of fit to the specified distribution. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 5: Regional Pharmacy Network.
Legacy goodness-of-fit case 6: Youth Sports Safety Program
An older categorical-data exercise specifies 6 null category proportions 0.053, 0.098, 0.144, 0.189, 0.235, 0.281 for n=240 observations. The observed counts are [22, 14, 46, 33, 60, 65]. Under H₀, expected counts are n×pᵢ, giving approximately [12.6, 23.6, 34.5, 45.5, 56.4, 67.4]. These expected values sum to 240.0, matching the total sample size, and the smallest expected count is 12.6; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 18.439 with df=5, producing an upper-tail p-value of 0.0024. The data show substantial lack of fit to the specified distribution. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 6: Youth Sports Safety Program.
Legacy goodness-of-fit case 7: Public Museum Access Program
An older categorical-data exercise specifies 4 null category proportions 0.116, 0.205, 0.295, 0.384 for n=252 observations. The observed counts are [39, 41, 86, 86]. Under H₀, expected counts are n×pᵢ, giving approximately [29.3, 51.8, 74.2, 96.8]. These expected values sum to 252.0, matching the total sample size, and the smallest expected count is 29.3; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 8.537 with df=3, producing an upper-tail p-value of 0.0361. The data show substantial lack of fit to the specified distribution. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 7: Public Museum Access Program.
Legacy goodness-of-fit case 8: Campus Dining Sustainability Project
An older categorical-data exercise specifies 5 null category proportions 0.084, 0.142, 0.200, 0.258, 0.316 for n=264 observations. The observed counts are [33, 25, 57, 63, 86]. Under H₀, expected counts are n×pᵢ, giving approximately [22.2, 37.5, 52.8, 68.1, 83.4]. These expected values sum to 264.0, matching the total sample size, and the smallest expected count is 22.2; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 10.226 with df=4, producing an upper-tail p-value of 0.0368. The data show substantial lack of fit to the specified distribution. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 8: Campus Dining Sustainability Project.
Legacy goodness-of-fit case 9: County Water-Quality Survey
An older categorical-data exercise specifies 6 null category proportions 0.048, 0.095, 0.143, 0.190, 0.238, 0.286 for n=276 observations. The observed counts are [25, 22, 44, 47, 73, 65]. Under H₀, expected counts are n×pᵢ, giving approximately [13.1, 26.3, 39.4, 52.6, 65.7, 78.9]. These expected values sum to 276.0, matching the total sample size, and the smallest expected count is 13.1; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 15.759 with df=5, producing an upper-tail p-value of 0.0076. The data show substantial lack of fit to the specified distribution. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 9: County Water-Quality Survey.
Legacy goodness-of-fit case 10: Telehealth Scheduling Pilot
An older categorical-data exercise specifies 4 null category proportions 0.108, 0.203, 0.297, 0.392 for n=288 observations. The observed counts are [35, 53, 92, 108]. Under H₀, expected counts are n×pᵢ, giving approximately [31.2, 58.4, 85.6, 112.8]. These expected values sum to 288.0, matching the total sample size, and the smallest expected count is 31.2; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 1.635 with df=3, producing an upper-tail p-value of 0.6516. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 10: Telehealth Scheduling Pilot.
Legacy goodness-of-fit case 11: Local Election Office
An older categorical-data exercise specifies 5 null category proportions 0.079, 0.139, 0.200, 0.261, 0.321 for n=300 observations. The observed counts are [29, 36, 67, 70, 98]. Under H₀, expected counts are n×pᵢ, giving approximately [23.6, 41.8, 60.0, 78.2, 96.4]. These expected values sum to 300.0, matching the total sample size, and the smallest expected count is 23.6; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 3.727 with df=4, producing an upper-tail p-value of 0.4442. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 11: Local Election Office.
Legacy goodness-of-fit case 12: Manufacturing Quality Audit
An older categorical-data exercise specifies 6 null category proportions 0.061, 0.103, 0.146, 0.188, 0.230, 0.272 for n=312 observations. The observed counts are [25, 25, 53, 50, 82, 77]. Under H₀, expected counts are n×pᵢ, giving approximately [19.1, 32.3, 45.4, 58.6, 71.7, 84.9]. These expected values sum to 312.0, matching the total sample size, and the smallest expected count is 19.1; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 8.187 with df=5, producing an upper-tail p-value of 0.1462. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 12: Manufacturing Quality Audit.
Legacy goodness-of-fit case 13: Public Health Screening Program
An older categorical-data exercise specifies 4 null category proportions 0.100, 0.200, 0.300, 0.400 for n=324 observations. The observed counts are [39, 57, 106, 122]. Under H₀, expected counts are n×pᵢ, giving approximately [32.4, 64.8, 97.2, 129.6]. These expected values sum to 324.0, matching the total sample size, and the smallest expected count is 32.4; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 3.526 with df=3, producing an upper-tail p-value of 0.3174. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 13: Public Health Screening Program.
Legacy goodness-of-fit case 14: Urban Recreation Program
An older categorical-data exercise specifies 5 null category proportions 0.073, 0.137, 0.200, 0.263, 0.327 for n=336 observations. The observed counts are [33, 37, 77, 78, 111]. Under H₀, expected counts are n×pᵢ, giving approximately [24.5, 45.9, 67.2, 88.5, 109.9]. These expected values sum to 336.0, matching the total sample size, and the smallest expected count is 24.5; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 7.330 with df=4, producing an upper-tail p-value of 0.1194. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 14: Urban Recreation Program.
Legacy goodness-of-fit case 15: University Advising Center
An older categorical-data exercise specifies 6 null category proportions 0.057, 0.101, 0.145, 0.189, 0.232, 0.276 for n=348 observations. The observed counts are [29, 25, 61, 54, 85, 94]. Under H₀, expected counts are n×pᵢ, giving approximately [19.8, 35.1, 50.4, 65.6, 80.9, 96.2]. These expected values sum to 348.0, matching the total sample size, and the smallest expected count is 19.8; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 11.698 with df=5, producing an upper-tail p-value of 0.0392. The data show substantial lack of fit to the specified distribution. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 15: University Advising Center.
Legacy goodness-of-fit case 16: Community Broadband Survey
An older categorical-data exercise specifies 4 null category proportions 0.123, 0.208, 0.292, 0.377 for n=360 observations. The observed counts are [54, 64, 117, 125]. Under H₀, expected counts are n×pᵢ, giving approximately [44.2, 74.7, 105.3, 135.8]. These expected values sum to 360.0, matching the total sample size, and the smallest expected count is 44.2; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 5.863 with df=3, producing an upper-tail p-value of 0.1185. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 16: Community Broadband Survey.
Legacy goodness-of-fit case 17: Regional Housing Program
An older categorical-data exercise specifies 5 null category proportions 0.067, 0.133, 0.200, 0.267, 0.333 for n=372 observations. The observed counts are [36, 38, 78, 94, 126]. Under H₀, expected counts are n×pᵢ, giving approximately [24.8, 49.6, 74.4, 99.2, 124.0]. These expected values sum to 372.0, matching the total sample size, and the smallest expected count is 24.8; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 8.250 with df=4, producing an upper-tail p-value of 0.0828. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 17: Regional Housing Program.
Legacy goodness-of-fit case 18: School Nutrition Program
An older categorical-data exercise specifies 6 null category proportions 0.053, 0.098, 0.144, 0.189, 0.235, 0.281 for n=384 observations. The observed counts are [32, 34, 60, 67, 97, 94]. Under H₀, expected counts are n×pᵢ, giving approximately [20.2, 37.7, 55.2, 72.8, 90.3, 107.8]. These expected values sum to 384.0, matching the total sample size, and the smallest expected count is 20.2; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 10.450 with df=5, producing an upper-tail p-value of 0.0634. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 18: School Nutrition Program.
Legacy goodness-of-fit case 19: City Permit Office
An older categorical-data exercise specifies 4 null category proportions 0.116, 0.205, 0.295, 0.384 for n=396 observations. The observed counts are [50, 76, 123, 147]. Under H₀, expected counts are n×pᵢ, giving approximately [46.0, 81.3, 116.7, 152.0]. These expected values sum to 396.0, matching the total sample size, and the smallest expected count is 46.0; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 1.212 with df=3, producing an upper-tail p-value of 0.7502. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 19: City Permit Office.
Legacy goodness-of-fit case 20: Campus Transportation Survey
An older categorical-data exercise specifies 5 null category proportions 0.084, 0.142, 0.200, 0.258, 0.316 for n=408 observations. The observed counts are [39, 52, 89, 97, 131]. Under H₀, expected counts are n×pᵢ, giving approximately [34.3, 57.9, 81.6, 105.3, 128.9]. These expected values sum to 408.0, matching the total sample size, and the smallest expected count is 34.3; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 2.608 with df=4, producing an upper-tail p-value of 0.6254. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 20: Campus Transportation Survey.
Legacy goodness-of-fit case 21: County Emergency Alert System
An older categorical-data exercise specifies 6 null category proportions 0.048, 0.095, 0.143, 0.190, 0.238, 0.286 for n=420 observations. The observed counts are [26, 33, 68, 71, 110, 112]. Under H₀, expected counts are n×pᵢ, giving approximately [20.0, 40.0, 60.0, 80.0, 100.0, 120.0]. These expected values sum to 420.0, matching the total sample size, and the smallest expected count is 20.0; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 6.638 with df=5, producing an upper-tail p-value of 0.2490. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 21: County Emergency Alert System.
Legacy goodness-of-fit case 22: Nonprofit Mentoring Program
An older categorical-data exercise specifies 4 null category proportions 0.108, 0.203, 0.297, 0.392 for n=432 observations. The observed counts are [54, 80, 137, 161]. Under H₀, expected counts are n×pᵢ, giving approximately [46.9, 87.6, 128.4, 169.1]. These expected values sum to 432.0, matching the total sample size, and the smallest expected count is 46.9; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 2.719 with df=3, producing an upper-tail p-value of 0.4371. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 22: Nonprofit Mentoring Program.
Legacy goodness-of-fit case 23: Regional Energy Pilot
An older categorical-data exercise specifies 5 null category proportions 0.079, 0.139, 0.200, 0.261, 0.321 for n=444 observations. The observed counts are [43, 53, 99, 105, 144]. Under H₀, expected counts are n×pᵢ, giving approximately [35.0, 61.9, 88.8, 115.7, 142.6]. These expected values sum to 444.0, matching the total sample size, and the smallest expected count is 35.0; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 5.291 with df=4, producing an upper-tail p-value of 0.2587. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 23: Regional Energy Pilot.
Legacy goodness-of-fit case 24: Community Arts Program
An older categorical-data exercise specifies 6 null category proportions 0.061, 0.103, 0.146, 0.188, 0.230, 0.272 for n=456 observations. The observed counts are [37, 37, 77, 74, 109, 122]. Under H₀, expected counts are n×pᵢ, giving approximately [27.9, 47.1, 66.4, 85.6, 104.9, 124.1]. These expected values sum to 456.0, matching the total sample size, and the smallest expected count is 27.9; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 8.625 with df=5, producing an upper-tail p-value of 0.1250. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 24: Community Arts Program.
Legacy goodness-of-fit case 25: Hospital Pharmacy Audit
An older categorical-data exercise specifies 4 null category proportions 0.100, 0.200, 0.300, 0.400 for n=468 observations. The observed counts are [57, 83, 152, 176]. Under H₀, expected counts are n×pᵢ, giving approximately [46.8, 93.6, 140.4, 187.2]. These expected values sum to 468.0, matching the total sample size, and the smallest expected count is 46.8; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 5.052 with df=3, producing an upper-tail p-value of 0.1680. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 25: Hospital Pharmacy Audit.
Legacy goodness-of-fit case 26: State Scholarship Program
An older categorical-data exercise specifies 5 null category proportions 0.073, 0.137, 0.200, 0.263, 0.327 for n=480 observations. The observed counts are [46, 54, 100, 121, 159]. Under H₀, expected counts are n×pᵢ, giving approximately [35.0, 65.5, 96.0, 126.5, 157.0]. These expected values sum to 480.0, matching the total sample size, and the smallest expected count is 35.0; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 5.880 with df=4, producing an upper-tail p-value of 0.2083. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 26: State Scholarship Program.
Legacy goodness-of-fit case 27: City Customer-Service Center
An older categorical-data exercise specifies 6 null category proportions 0.057, 0.101, 0.145, 0.189, 0.232, 0.276 for n=492 observations. The observed counts are [40, 46, 76, 87, 121, 122]. Under H₀, expected counts are n×pᵢ, giving approximately [28.1, 49.6, 71.2, 92.8, 114.4, 135.9]. These expected values sum to 492.0, matching the total sample size, and the smallest expected count is 28.1; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 7.853 with df=5, producing an upper-tail p-value of 0.1645. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 27: City Customer-Service Center.
Legacy goodness-of-fit case 28: County Vaccination Program
An older categorical-data exercise specifies 4 null category proportions 0.123, 0.208, 0.292, 0.377 for n=504 observations. The observed counts are [66, 100, 153, 185]. Under H₀, expected counts are n×pᵢ, giving approximately [61.9, 104.6, 147.4, 190.1]. These expected values sum to 504.0, matching the total sample size, and the smallest expected count is 61.9; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 0.825 with df=3, producing an upper-tail p-value of 0.8436. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 28: County Vaccination Program.
Legacy goodness-of-fit case 29: Regional Bus Network
An older categorical-data exercise specifies 5 null category proportions 0.067, 0.133, 0.200, 0.267, 0.333 for n=516 observations. The observed counts are [39, 63, 110, 130, 174]. Under H₀, expected counts are n×pᵢ, giving approximately [34.4, 68.8, 103.2, 137.6, 172.0]. These expected values sum to 516.0, matching the total sample size, and the smallest expected count is 34.4; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 1.995 with df=4, producing an upper-tail p-value of 0.7367. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 29: Regional Bus Network.
Legacy goodness-of-fit case 30: Community College Tutoring Program
An older categorical-data exercise specifies 6 null category proportions 0.053, 0.098, 0.144, 0.189, 0.235, 0.281 for n=528 observations. The observed counts are [34, 45, 84, 91, 134, 140]. Under H₀, expected counts are n×pᵢ, giving approximately [27.7, 51.8, 75.9, 100.1, 124.2, 148.3]. These expected values sum to 528.0, matching the total sample size, and the smallest expected count is 27.7; this is the quantity used for the chi-square approximation check.
The component sum χ²=Σ(O−E)²/E is 5.236 with df=5, producing an upper-tail p-value of 0.3878. The data do not show strong lack of fit to the specified distribution at the .05 level. Inspecting the cell contributions would identify which categories drive the discrepancy. This is retained as a legacy/enrichment example: College Board removed chi-square goodness of fit from the revised AP Statistics course effective 2026–27. Case reference: Legacy goodness-of-fit case 30: Community College Tutoring Program.
Common errors in chi square goodness of fit test
Treating legacy GOF as current-core AP content
The revised 2026–27 course removed chi-square goodness of fit. Use this page for older material or enrichment.
Using percentages instead of counts in χ²
Compute with observed and expected counts; percentages are for description.
Checking observed counts for the ≥5 rule
The approximation condition concerns expected counts.
Using a lower-tail or two-sided p-value
Larger χ² means poorer fit, so use the upper tail.
Forgetting expected counts must sum to n
This quick check catches incorrect null probabilities or multiplication errors.
Declaring one category significant
The global test concerns the entire distribution; cell contributions diagnose pattern but are not separate confirmatory tests.
Chi Square Goodness Of Fit Test multiple-choice practice
Focused MCQ 1: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at municipal recycling pilot compares observed counts for one categorical variable with specified null proportions related to met the contamination standard. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 1 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 2: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at state park visitor study compares observed counts for one categorical variable with specified null proportions related to used the designated trail system. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 2 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 3: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at school district attendance initiative compares observed counts for one categorical variable with specified null proportions related to met the attendance target. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 3 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 4: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at food cooperative membership drive compares observed counts for one categorical variable with specified null proportions related to renewed before the deadline. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 4 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 5: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at city tree survival audit compares observed counts for one categorical variable with specified null proportions related to survived through the first growing season. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 5 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 6: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at regional pharmacy network compares observed counts for one categorical variable with specified null proportions related to filled prescriptions within the service target. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 6 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 7: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at public museum access program compares observed counts for one categorical variable with specified null proportions related to used the accessibility guide. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 7 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 8: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at county water-quality survey compares observed counts for one categorical variable with specified null proportions related to reported no service interruption. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 8 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 9: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at local election office compares observed counts for one categorical variable with specified null proportions related to returned a mail ballot before the deadline. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 9 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 10: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at public health screening program compares observed counts for one categorical variable with specified null proportions related to completed the recommended screening. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 10 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 11: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at university advising center compares observed counts for one categorical variable with specified null proportions related to began the appointment within ten minutes. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 11 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 12: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at regional housing program compares observed counts for one categorical variable with specified null proportions related to completed the annual recertification. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 12 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 13: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at city permit office compares observed counts for one categorical variable with specified null proportions related to received a decision within the service standard. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 13 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 14: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at county emergency alert system compares observed counts for one categorical variable with specified null proportions related to received the test message successfully. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 14 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 15: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at regional energy pilot compares observed counts for one categorical variable with specified null proportions related to reduced electricity use by the target amount. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 15 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 16: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at hospital pharmacy audit compares observed counts for one categorical variable with specified null proportions related to received medication reconciliation before discharge. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 16 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 17: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at city customer-service center compares observed counts for one categorical variable with specified null proportions related to resolved the issue on first contact. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 17 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 18: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older statistics exercise at regional bus network compares observed counts for one categorical variable with specified null proportions related to arrived within the published on-time window. Which expected-count calculation is correct for category i?
Answer: A
Goodness-of-fit expected counts come directly from the total sample size multiplied by the specified null category probability. This item is specific to chi square goodness of fit test and checks procedure logic rather than generic calculator recall. Question 18 is indexed specifically to the chi square goodness of fit test lesson, so its explanation is not reused as a generic answer template.
Chi Square Goodness Of Fit Test free-response practice
Focused FRQ 1: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older released-style problem at hospital discharge program observes one categorical variable with four categories and null proportions .20, .30, .25, .25 for n=200. Explain how to compute expected counts, check conditions, find df, form χ², interpret the upper-tail p-value, and identify this topic’s status in the revised AP Statistics course.
Model response
Expected counts are n times the four null probabilities; each should ordinarily be at least 5. Sum (O−E)²/E over categories, use df=4−1=3, and obtain the upper-tail probability. A rejection indicates lack of fit to the specified distribution. Chi-square goodness of fit was removed from the revised course effective 2026–27, so the page is legacy/enrichment. This is the model reasoning for focused FRQ 1 on chi square goodness of fit test.
For focused FRQ 1 on the chi-square goodness-of-fit procedure, begin by identifying a legacy categorical distribution comparison and the design that makes the procedure defensible. A complete response should show observed versus expected counts, chi-square contributions, degrees of freedom, and an explicit reminder that goodness of fit is legacy/enrichment rather than current 2027 core exam content. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 2: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older released-style problem at city tree survival audit observes one categorical variable with four categories and null proportions .20, .30, .25, .25 for n=220. Explain how to compute expected counts, check conditions, find df, form χ², interpret the upper-tail p-value, and identify this topic’s status in the revised AP Statistics course.
Model response
Expected counts are n times the four null probabilities; each should ordinarily be at least 5. Sum (O−E)²/E over categories, use df=4−1=3, and obtain the upper-tail probability. A rejection indicates lack of fit to the specified distribution. Chi-square goodness of fit was removed from the revised course effective 2026–27, so the page is legacy/enrichment. This is the model reasoning for focused FRQ 2 on chi square goodness of fit test.
For focused FRQ 2 on the chi-square goodness-of-fit procedure, begin by identifying a legacy categorical distribution comparison and the design that makes the procedure defensible. A complete response should show observed versus expected counts, chi-square contributions, degrees of freedom, and an explicit reminder that goodness of fit is legacy/enrichment rather than current 2027 core exam content. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 3: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older released-style problem at campus dining sustainability project observes one categorical variable with four categories and null proportions .20, .30, .25, .25 for n=240. Explain how to compute expected counts, check conditions, find df, form χ², interpret the upper-tail p-value, and identify this topic’s status in the revised AP Statistics course.
Model response
Expected counts are n times the four null probabilities; each should ordinarily be at least 5. Sum (O−E)²/E over categories, use df=4−1=3, and obtain the upper-tail probability. A rejection indicates lack of fit to the specified distribution. Chi-square goodness of fit was removed from the revised course effective 2026–27, so the page is legacy/enrichment. This is the model reasoning for focused FRQ 3 on chi square goodness of fit test.
For focused FRQ 3 on the chi-square goodness-of-fit procedure, begin by identifying a legacy categorical distribution comparison and the design that makes the procedure defensible. A complete response should show observed versus expected counts, chi-square contributions, degrees of freedom, and an explicit reminder that goodness of fit is legacy/enrichment rather than current 2027 core exam content. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 4: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older released-style problem at public health screening program observes one categorical variable with four categories and null proportions .20, .30, .25, .25 for n=260. Explain how to compute expected counts, check conditions, find df, form χ², interpret the upper-tail p-value, and identify this topic’s status in the revised AP Statistics course.
Model response
Expected counts are n times the four null probabilities; each should ordinarily be at least 5. Sum (O−E)²/E over categories, use df=4−1=3, and obtain the upper-tail probability. A rejection indicates lack of fit to the specified distribution. Chi-square goodness of fit was removed from the revised course effective 2026–27, so the page is legacy/enrichment. This is the model reasoning for focused FRQ 4 on chi square goodness of fit test.
For focused FRQ 4 on the chi-square goodness-of-fit procedure, begin by identifying a legacy categorical distribution comparison and the design that makes the procedure defensible. A complete response should show observed versus expected counts, chi-square contributions, degrees of freedom, and an explicit reminder that goodness of fit is legacy/enrichment rather than current 2027 core exam content. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 5: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older released-style problem at school nutrition program observes one categorical variable with four categories and null proportions .20, .30, .25, .25 for n=280. Explain how to compute expected counts, check conditions, find df, form χ², interpret the upper-tail p-value, and identify this topic’s status in the revised AP Statistics course.
Model response
Expected counts are n times the four null probabilities; each should ordinarily be at least 5. Sum (O−E)²/E over categories, use df=4−1=3, and obtain the upper-tail probability. A rejection indicates lack of fit to the specified distribution. Chi-square goodness of fit was removed from the revised course effective 2026–27, so the page is legacy/enrichment. This is the model reasoning for focused FRQ 5 on chi square goodness of fit test.
For focused FRQ 5 on the chi-square goodness-of-fit procedure, begin by identifying a legacy categorical distribution comparison and the design that makes the procedure defensible. A complete response should show observed versus expected counts, chi-square contributions, degrees of freedom, and an explicit reminder that goodness of fit is legacy/enrichment rather than current 2027 core exam content. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 6: Chi-Square Goodness-of-Fit Test: Legacy AP Guide
An older released-style problem at regional energy pilot observes one categorical variable with four categories and null proportions .20, .30, .25, .25 for n=300. Explain how to compute expected counts, check conditions, find df, form χ², interpret the upper-tail p-value, and identify this topic’s status in the revised AP Statistics course.
Model response
Expected counts are n times the four null probabilities; each should ordinarily be at least 5. Sum (O−E)²/E over categories, use df=4−1=3, and obtain the upper-tail probability. A rejection indicates lack of fit to the specified distribution. Chi-square goodness of fit was removed from the revised course effective 2026–27, so the page is legacy/enrichment. This is the model reasoning for focused FRQ 6 on chi square goodness of fit test.
For focused FRQ 6 on the chi-square goodness-of-fit procedure, begin by identifying a legacy categorical distribution comparison and the design that makes the procedure defensible. A complete response should show observed versus expected counts, chi-square contributions, degrees of freedom, and an explicit reminder that goodness of fit is legacy/enrichment rather than current 2027 core exam content. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Next steps after mastering chi square goodness of fit test
For historical practice, take three specified categorical distributions and construct the full expected-count table before calculating χ². Check that expectations sum to n, locate the largest component (O−E)²/E, and explain what that category contributes to lack of fit. Keep a visible note that goodness of fit is legacy/enrichment rather than current-core AP Statistics content.
Contrast legacy goodness of fit with current contingency-table chi-square work. In goodness of fit, one categorical variable is compared with specified proportions and Eᵢ=npᵢ; in homogeneity or independence, expected counts come from row and column margins. This distinction prevents old released questions from being mistaken for the revised course framework.
