2019 ASRJC P2 Q6

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 in the table below.

1. 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.[4]

Studies also show that babies born to mothers who smoke have a lower mean birth weight of \({2.80}\) kg.

For the remaining of the question, you may assume the birth weight of babies born to mothers who smoke is also normally distributed and that the standard deviations for the birth weight of babies born to mothers who do not smoke and mothers who smoke are both \({0.86}\) kg.

2. Three pregnant mothers-to-be who smoke are randomly chosen. Find the probability that all their babies will not be of normal birth weight. State an assumption that is needed for your working.[3]
3. Find the probability that the average birth weight of two babies from mothers who do not smoke differs from twice the birth weight of a baby from a mother who smokes by less than \({2}\) kg.[3]

Babies whose birth weight not classified as normal will have to remain in hospital for further observation until their condition stabilises. Depending on the treatment received and length of stay, the mean hospitalisation cost per baby is \({\$2800}\) and standard deviation is \({\$500}\).

4. Find the probability that the average hospitalisation cost from a random sample of \({50}\) babies, whose birth weight not classified as normal, exceeds \({\$3000}\).[2]