2023 NJC P2 Q9

It is believed that the probability \(p\) of a randomly chosen pregnant woman giving birth to a Down's Syndrome child is related to the woman’s age \(x\), in years. The table gives the observed values of \(p\) for 5 different values of \(x\).

1. Sketch the scatter diagram representing the data and comment if \(p=mx+c\) is a good model for this set of data.[2]
2. By finding the product-moment correlation coefficient for each of the following models, determine which of the following would be an appropriate model to represent the above data:
   A: \(p=a+\frac{b}{x}\)
   B: \(p=a+b{{e}^{-x}}\)
   C: \(\ln p=a+bx\),
   where \(a\) and \(b\) are constants.[3]
3. For the appropriate model, calculate the values of \(a\) and \(b\).[2]
4. Obtain an estimate of the probability of a \(19\)-year-old woman giving birth to a Down’s Syndrome child. Comment on the reliability of your answer.[2]