- A doctor is studying the scans of 30 -week old foetuses. She takes a random sample of 8 scans and measures the length, \(f \mathrm {~mm}\), of the leg bone called the femur. She obtains the following results.
$$\begin{array} { l l l l l l l l }
52 & 53 & 56 & 57 & 57 & 59 & 60 & 62
\end{array}$$
- Show that \(\mathrm { S } _ { f f } = 80\)
The doctor also measures the head circumference, \(h \mathrm {~mm}\), of each foetus and her results are summarised as
$$\sum h = 2209 \quad \sum h ^ { 2 } = 610463 \quad \mathrm {~S} _ { f h } = 182$$
- Find \(\mathrm { S } _ { h h }\)
- Calculate the product moment correlation coefficient between the length of the femur and the head circumference for these data.
The doctor believes that there is a linear relationship between the length of the femur and the head circumference of 30-week old foetuses.
- State, giving a reason, whether or not your calculation in part (c) supports the doctor's belief.
- Find an equation of the regression line of \(h\) on \(f\).
The doctor plans in future to measure the femur length, \(f\), and then use the regression line to estimate the corresponding head circumference, \(h\).
A statistician points out that there will always be the chance of an error between the true head circumference and the estimated value of the head circumference.
Given that the error, \(E \mathrm {~mm}\), has the normal distribution \(\mathrm { N } \left( 0,4 ^ { 2 } \right)\)
- find the probability that the estimate is within 3 mm of the true value.