Model Answer
0 min readIntroduction
The question presents a classic scenario involving statistical inference and hypothesis testing. It asks us to evaluate whether observed data supports a claim about the relationship between two variables: a father’s skill level and his son’s intelligence. This type of analysis is fundamental in various fields, including social sciences, education, and public policy, where understanding correlations between different factors is essential for informed decision-making. The core of the problem lies in determining if the association between skilled fathers and intelligent sons is statistically significant or merely due to chance.
Organizing the Data
First, let's organize the given information into a contingency table for clarity:
| Intelligent Boys | Unintelligent Boys | Total | |
|---|---|---|---|
| Skilled Fathers | 40 | (75-40) = 35 | 75 |
| Unskilled Fathers | (200-75) - 85 = 40 | 85 | 125 |
| Total | 80 | 120 | 200 |
Calculating Probabilities
To assess the hypothesis, we need to calculate the following conditional probabilities:
- P(Intelligent | Skilled Father): The probability that a boy is intelligent given that his father is skilled.
- P(Intelligent | Unskilled Father): The probability that a boy is intelligent given that his father is unskilled.
Using the data from the table:
- P(Intelligent | Skilled Father) = 40 / 75 = 0.5333 (approximately)
- P(Intelligent | Unskilled Father) = 40 / 125 = 0.32
Interpreting the Results
Comparing the two probabilities, we observe that the probability of a boy being intelligent is significantly higher (0.5333) if his father is skilled compared to when his father is unskilled (0.32). This suggests a positive correlation between a father’s skill level and his son’s intelligence.
Hypothesis Testing (Informal)
While a formal hypothesis test (like a chi-squared test) isn't explicitly required given the question's scope, we can informally assess the strength of the evidence. The substantial difference in probabilities (0.5333 vs. 0.32) indicates that the observed data is not easily explained by random chance. A skilled father is approximately 66.67% more likely to have an intelligent son than an unskilled father (0.5333/0.32 = 1.6667).
Addressing the Hypothesis
The figures do support the hypothesis that skilled fathers have intelligent boys. The calculated probabilities demonstrate a clear association between the two variables. However, it's crucial to remember that correlation does not equal causation. Other factors, such as socioeconomic status, access to education, and genetic predisposition, could also contribute to a child’s intelligence.
Limitations
The analysis is based on a limited dataset of 200 boys. A larger sample size would provide more robust evidence. Additionally, the definition of "intelligent" and "skilled" is not provided, which could introduce ambiguity. The study doesn't account for the mother's influence or other environmental factors.
Conclusion
In conclusion, the provided data strongly suggests a positive correlation between having a skilled father and having an intelligent son. The calculated probabilities demonstrate that boys with skilled fathers are significantly more likely to be intelligent than those with unskilled fathers. While this supports the hypothesis, it’s important to acknowledge that correlation doesn’t imply causation and other factors likely play a role. Further research with a larger sample size and more defined variables would be needed to establish a more definitive relationship.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.