Does industrial development of a country affect average lifespan? I have collected data on 15 randomly selected countries. Five of each from low, middle, and high levels of industrial development. The statistics and box plots are below. Are the conditions for one-factor ANOVA met?

Income
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1 high 5 76.4 5.59
2 low 5 57 3.16
3 middle 5 75.6 4.39

Question

Does industrial development of a country affect average lifespan? I have collected data on 15 randomly selected countries. Five of each from low, middle, and high levels of industrial development. The statistics and box plots are below. Are the conditions for one-factor ANOVA met?

Income   
𝑛
(
)
  
𝑚
𝑒
𝑎
𝑛
(
𝐿
if
𝑒
)
  
𝑠
𝑑
(
𝐿
if
𝑒
)

1 high         5         76.4                5.59
2 low          5         57                   3.16
3 middle     5         75.6                4.39

BlackTom AI · Accepted Answer

For one-way ANOVA the key assumptions are independence of observations, approximate normality of the response within each group, and (approximate) equality of group variances (homoscedasticity).

1) Relevant knowledge points and concepts
- Independence: observations should be randomly selected and groups independent of each other.  
- Normality: each group’s sampling distribution should be roughly normal; ANOVA is robust to modest departures when samples are balanced.  
- Equal variances: a common rule-of-thumb is that the largest group standard deviation should not exceed about twice the smallest (and balanced designs tolerate more deviation).

2) Step-by-step reasoning
- Independence: the problem states 15 randomly selected countries with five from each income level, so groups can be treated as independent random samples.  
- Normality: with n = 5 per group we cannot rigorously test normality, but the provided boxplots show no extreme skewness or outliers in any group; distributions look roughly symmetric. Balanced samples (equal n) make ANOVA robust to modest nonnormality.  
- Equal variances: use the reported standard deviations: sd_high = 5.59, sd_low = 3.16, sd_middle = 4.39. Compute ratios to check homoscedasticity.

3) Intermediate calculations
- largest/smallest sd = 5.59 / 3.16 = 1.77
- other comparisons: 5.59 / 4.39 = 1.27; 4.39 / 3.16 = 1.39

4) Why the correct answer is correct
- All three assumptions are reasonably satisfied: random independent sampling, no visible departures from normality in boxplots, and variance ratios well below the common cutoff of 2 (1.77 here). The balanced design (n = 5 in each group) increases ANOVA’s robustness to small departures. Therefore the conditions for one-factor ANOVA are met.

5) Why other choices are incorrect
- “No, some part of the binomial condition is not met.” — The binomial condition is irrelevant to ANOVA (it applies to proportion problems).  
- “No standard deviations are too different” — the sd ratio is < 2, so variances are acceptably similar.  
- “No, normality cannot be met.” — while small samples limit formal testing, the boxplots show no extreme nonnormality and ANOVA is robust for balanced samples.

6) Conclusion
All ANOVA conditions are met; this agrees with the correct answer: Yes all conditions are met.

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