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 women's age $x$, in years. The table gives the observed values of $p$ for 5 different values of $x$.

$x$	25	30	35	40	45
$p$	0.00067	0.00125	0.00333	0.01000	0.03330

1. Sketch the scatter diagram for the given data

[Sketch of a scatter diagram plotting the given points $(x, p)$ with $x$ on the horizontal axis and $p$ on the vertical axis. The points (25, 0.00067), (30, 0.00125), (35, 0.00333), (40, 0.01000) and (45, 0.03330) are plotted, clearly showing an upward-bending curve.]

2. Comment on the appropriateness of the linear model

Since the points on the scatter diagram exhibit a non-linear trend and follow a curve rather than a straight line, the linear model $p = mx + c$ is not a good model for this set of data.

1. Calculate the product-moment correlation coefficient for Model A

For Model A ($p = a + \frac{b}{x}$), the linear relationship is between $\frac{1}{x}$ and $p$.
By entering the data pairs $\left(\frac{1}{x}, p\right)$ into a graphing calculator, we obtain the product-moment correlation coefficient:
$r \approx -0.7628$

2. Calculate the product-moment correlation coefficient for Model B

For Model B ($p = a + b e^{-x}$), the linear relationship is between $e^{-x}$ and $p$.
Entering the data pairs $(e^{-x}, p)$ into the graphing calculator gives:
$r \approx -0.3719$

3. Calculate the product-moment correlation coefficient for Model C

For Model C ($\ln p = a + bx$), the relationship is directly linear between $x$ and $\ln p$.
Entering the data pairs $(x, \ln p)$ into the graphing calculator yields:
$r \approx 0.9937$

4. Determine the most appropriate model based on the correlation coefficients

To determine the best fit, we compare the absolute values of the correlation coefficients ($|r|$).
Model C has a correlation coefficient magnitude closest to $1$ ($|0.9937| \approx 1$), which indicates the strongest linear correlation among the three models.

Thus, Model C is the most appropriate model.

1. Identify the appropriate model to use based on the expected results from part (ii).

From part (ii), Model C ($\ln p = a + bx$) yields a product-moment correlation coefficient closest to $1$ ($r \approx 0.994$), making it the most appropriate model for the data.

2. Use a graphing calculator to compute the linear regression coefficients for the chosen model.

By entering the values of $x$ and $\ln p$ into a graphing calculator and performing a linear regression of $\ln p$ on $x$:
$a = -12.465 \approx -12.5$ (to 3 s.f.)
$b = 0.19783 \approx 0.198$ (to 3 s.f.)

1. Substitute $x = 19$ into the regression model for C to find the estimated probability.

From part (iii), the constants for Model C ($\ln p = a + bx$) are $a \approx -12.465$ and $b \approx 0.19783$.
Substituting $x = 19$,
$\ln p = -12.46508 + 0.1978308(19)$
$\ln p = -8.7063$
$p = e^{-8.7063} \approx 0.000166 \text{ (to 3 s.f.)}$

2. Comment on the reliability of the estimate.

The estimate is unreliable because $x = 19$ is outside the range of given data values ($25 \le x \le 45$), meaning this is an extrapolation.