---
title: "Chi-square: intelligence vs economic condition (1650)"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:chi-square-intelligence-economic-1650"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/chi-square-intelligence-economic-1650
site: BCT Engineering
---

# Chi-square: intelligence vs economic condition (1650)

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2076 Chaitra, 5 marks
- Video: [Chi Square Test](https://www.youtube.com/watch?v=HKDqlYSLt68) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/chi-square-intelligence-economic-1650
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

In a recent survey 1650 Engineers were classified according to their intelligence (GPA in Bachelor) and economic condition after graduation. Test whether there is any association between intelligence and economic condition. [χ² value for 9 degree of freedom = 16.919]
| Economic Condition after graduation | Excellent | Good | Mediocre | Dull |
|---|---|---|---|---|
| Good | 48 | 199 | 181 | 82 |
| Average | 80 | 73 | 65 | 89 |
| Below average | 86 | 51 | 51 | 84 |
| Not good | 81 | 185 | 190 | 105 |

## Short answer

Chi-square test yields χ²=149.513 with 9 d.f., critical value 16.919 at 5%. Reject H0; there is a highly significant association between intelligence and economic condition.

## Answer

Chi-square test of independence for the 4 × 4 table. Grand total = 1650.

1. H0: the two attributes are independent (no association).
   H1: the two attributes are associated (dependent).
2. Level of significance: α = 0.05
3. Test statistic: χ² = Σ (O − E)²/E, where E = (row total × column total)/grand total.

Observed frequencies with totals:

```text
               Excellent      Good  Mediocre      Dull     Total
Good                  48       199       181        82       510
Average               80        73        65        89       307
Below avg             86        51        51        84       272
Not good              81       185       190       105       561
Total                295       508       487       360      1650
```

Expected frequencies:

```text
               Excellent      Good  Mediocre      Dull
Good               91.18    157.02    150.53    111.27
Average            54.89     94.52     90.61     66.98
Below avg          48.63     83.74     80.28     59.35
Not good          100.30    172.72    165.58    122.40
```

Terms (O − E)²/E:

```text
               Excellent      Good  Mediocre      Dull
Good              20.450    11.225     6.169     7.701
Average           11.489     4.899     7.239     7.238
Below avg         28.717    12.802    10.680    10.243
Not good           3.714     0.873     3.602     2.474
```

χ² = Σ (O − E)²/E = **149.513**

4. Degrees of freedom = (r − 1)(c − 1) = (4 − 1)(4 − 1) = **9**.
   Critical value: χ²(0.05; 9) = **16.919**. Reject H0 if χ² > 16.919.
5. Decision: calculated χ² > tabulated χ² (149.513 > 16.919, p-value < 0.0001), so **reject H0**.

**Conclusion:** there is a significant association between the two attributes; they are not independent.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are the d.f. for the 4 × 4 table?
   A. 16
   B. 12
   C. 3
   D. 9
   Answer: D. (4 − 1)(4 − 1) = 9, matching the given 16.919. (Key point: Degrees of freedom)

2. What is E for 'Good, Excellent'?
   A. 510 × 295/1650 = 91.18
   B. 48
   C. 510/4
   D. 295/4
   Answer: A. Row total × column total / grand total = 510 × 295/1650. (Key point: Expected frequency)

3. What does H0 state?
   A. Intelligence and economic condition are associated
   B. The attributes are independent
   C. All frequencies are equal
   D. The means are equal
   Answer: B. H0 for a test of independence says no association. (Key point: Hypotheses)

4. Calculated chi-square is 149.513 vs 16.919. What is the conclusion?
   A. Cannot decide
   B. Accept H0
   C. Reject H0; significant association
   D. Independent
   Answer: C. 149.513 far exceeds 16.919, so independence is rejected. (Key point: Decision)

## Close variants set in other exams

- [Chi-square: intelligence vs economic condition (1072)](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Achi-square-intelligence-economic-1072) — Probability and Statistics, chapter 7, asked 3×

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