IB AA SL: core and optional content Core: individual-height probability in(b)(i) and qualitative mod

Question

IB AA SL: core and optional content

Core: individual-height probability in(b)(i) and qualitative model discussion in(a)(i)–(ii). Optional: distribution of a sample mean in(a)(iii), total-height probability in(b)(ii) and dependent comparison.

A school has a large number of girls. For each year-group it may be assumed that the heights of girls are normally distributed, with average heights and standard deviations as given in the following table.

Year-group

Average height

Standard deviation

$11$-year old girls

$1.1$ m

$0.2$ m

$16$-year old girls

$1.7$ m

$0.3$ m

For each of the following cases, state, with a reason, whether or not a normal model is likely to be appropriate.
1. The number of days since her last birthday of a randomly chosen $11$-year old girl from the school.
2. The height of a pupil chosen at random from the combined group of $11$-year old and $16$-year old girls in the school.
3. The average height of a randomly chosen sample of eight $11$-year old girls from the school
Find the probability such that
1. each of the two randomly chosen $11$-year old girls is taller than $1.2$ m.
2. the total height of two randomly chosen $11$-year old girls is more than $2.4$ m.

Explain why the answer to (b)(ii) is higher than that of (b)(i).

Accepted Answer

(a)(i) Not appropriate: days since the last birthday are approximately uniformly distributed. (a)(ii) Not appropriate: combining the two year-groups produces a mixture that is not exactly normal; bimodality depends on the mixing proportions. (a)(iii) Appropriate: $\overline X\sim N\left(1.1,\dfrac{0.2^2}{8}\right)$. (b)(i) $0.0952$ (b)(ii) $0.240$ Explanation: Event (b)(i) is a proper subset of event (b)(ii), so (b)(ii) has the greater probability.

Linear Combinations, Parameter Estimation, and Real-World Modeling

Linear Combinations, Parameter Estimation, and Real-World Modeling

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