A teacher took a random sample of 8 children from a class - Edexcel - A-Level Maths Statistics - Question 7 - 2011 - Paper 2

To find the regression line of $h$ on $f$, we need to calculate the slope $b$ and the intercept $a$:

1. Calculate the slope: $b = \frac{S_{fh}}{S_{ff}}$ Given that $S_{ff} = 39.5$, we find: $b = \frac{82.75}{39.5} \approx 2.09494 \approx 2.09 \text{ (to 3 significant figures)}$

2. Calculate the intercept: Using the formula: $a = \frac{\sum h}{n} - b \times \frac{\sum f}{n}$ We find: $a = \frac{1085}{8} - 2.09 \times \frac{186}{8}$ $a = 135.625 - 48.8075 \approx 86.82 \text{ (to 3 significant figures)}$

Therefore, the regression line is: $h = 86.82 + 2.09f$

Using the regression equation $h = 86.82 + 2.09f$, we can substitute $f = 25$:

$h = 86.82 + 2.09 \times 25$ $h = 86.82 + 52.25$ $h \approx 139.07 \text{ cm}$

Thus, the estimated height of a child with a left foot length of 25 cm is approximately 139.07 cm.

The estimate of the height of a child with a foot length of 25 cm can be considered reliable as long as this foot length falls within the range of the data collected. In this case, since the data only includes foot lengths from 20 cm to 27 cm, an estimate at 25 cm is well within this range, suggesting that the linear relationship established is appropriate for this estimate.

The equation derived in part (b) is based on children’s data, and using it to estimate the height of an adult, like the teacher, may not be valid as adults typically have different growth patterns and physical dimensions than children. This regression is specifically tailored to the sample population of children and is therefore unsuitable for extrapolation to adults.