Mathematics 9709/73 — October/November 2012

Walkthrough

We are told that each log length is normally distributed with mean $3.5$ m and standard deviation $0.12$ m. Choosing 8 logs randomly means we have 8 independent normal random variables. The total length is their sum. A key result for this topic is that a linear combination of independent normal variables is also normal, so the total length is normal. To describe the distribution fully, we need its mean and variance.

For a sum of independent variables, the mean is the sum of the means:

For independent variables, the variances add. Since the variance of one log is $0.12^2$, the total variance is:

So the total length is $N(28, 0.1152)$. To 3 significant figures, the variance is $0.115\,\text{m}^2$. Alternatively, the standard deviation is $\sqrt{0.1152} = 0.339$ m. The mark scheme accepts either clearly stated variance or standard deviation.

Key Takeaways

• The sum of independent normal random variables is normally distributed.
• For a sum, means add and variances add; standard deviations do not add.
• Always square the standard deviation before multiplying by the number of variables, then take the square root if a standard deviation is required.
• “Describe fully” means state the type of distribution and all its parameters (mean and variance or standard deviation).

Common Mistakes

• Adding standard deviations: $8 \times 0.12 = 0.96$ is incorrect because variances, not standard deviations, add.
• Forgetting to square $0.12$ before multiplying by 8.
• Giving only the mean or only the variance without stating that the distribution is normal.
• Not rounding to 3 significant figures, or giving the variance when the question asks for standard deviation without saying which one is being given.

Things to Be Careful About

• The logs must be independent for the variances to add; “randomly chosen” indicates independence.
• The variance has units $\text{m}^2$, while the standard deviation has units m.
• In the mark scheme, B1 is for “Normal with mean 28”, M1 is for $0.12^2 \times 8$, and A1 is for $0.115$ (3 s.f.). If you give the standard deviation $0.339$ instead, make sure it is clearly labelled as the standard deviation.