<p>Birth weight can be used to predict short and long-term health complications for babies. Studies show that the birth weight of babies born to mothers who do not smoke in a certain hospital can be assumed to follow a normal distribution with mean $3.05$ kg and variance ${{\sigma }^{2}}$kg$^{2}$.</p>
<p>The hospital classifies babies based on their birth weight as shown as in the table below.</p>
<table><thead><tr><th>Birth weight</th><th>Classification</th></tr></thead><tbody><tr><td>Less than $1.5$ kg</td><td>Very low birth weight</td></tr><tr><td>$1.5$ kg to $2.5$ kg</td><td>Low birth weight</td></tr><tr><td>$2.5$ kg to $4.0$ kg</td><td>Normal birth weight</td></tr><tr><td>More than $4.0$ kg</td><td>High birth weight</td></tr></tbody></table>
<ol class="i-list"><li><div class="list-item-paragraph">A sample showed that $20.2%$ of the babies born to mothers who do not smoke have low birth weight. If this is true for the entire population, find two possible values of $\sigma $, corrected to $2$ decimal places. Explain clearly why one of the values of $\sigma $ found should be rejected.</div><p style="text-align: right;">[4]</p></li></ol>

Birth weight can be used to predict short and long-term health complications for babies. Studies show that the birth weight of babies born to mothers who do not smoke in a certain hospital can be assumed to follow a normal distribution with mean $3.05$ kg and variance ${{\sigma }^{2}}$kg$^{2}$.

The hospital classifies babies based on their birth weight as shown as in the table below.

A sample showed that $20.2%$ of the babies born to mothers who do not smoke have low birth weight. If this is true for the entire population, find two possible values of $\sigma $, corrected to $2$ decimal places. Explain clearly why one of the values of $\sigma $ found should be rejected.

Normal Distribution - General

Normal Distribution

H2 Math

International Baccalaureate Higher Level Mathematics

IB HL Math

2019 ASRJC P2 Q6

Let X be the weight of a randomly baby born to mothers who do not smoke.

$X \sim N(3.05, \sigma^2)$

$P(1.5 < X < 2.5) = 0.202$

$P(1.5 < X < 2.5) - 0.202 = 0$

Using GC: Y1=normalcdf(1.5,2.5,3.05,X)-0.202
Zero: X=1.5599483

From GC,
$\sigma = 0.697$ (3s.f.) or $1.56$ (3s.f.)
$\sigma = 0.70$ (2d.p.) or $1.56$ (2d.p.)

$X \sim N(3.05, 0.70^2)$ vs $X \sim N(3.05, 1.56^2)$

when $\sigma = 0.70$, $P(X < 0) = 6.59 \times 10^{-6} = 0.000000657 \approx 0$

$\sigma = 1.56$, $P(X < 0) = 0.02528 \approx 2.5\%$

I.e. On average, for every 100 babies born, 2.5 babies have a weight less than 0 kg.
$\therefore \sigma = 1.56$ should be rejected.

2019 ASRJC P2 Q6

2019 ASRJC P2 Q6

Video Solution: