一位教师对7名学生进行了调查，以研究每周“屏幕时间”小时数（$h$）与他们在最近结束的考试中获得的平均分数（$s$）之间的关系。她将自己的调查结果记录在下表中。

1. Find the mean of the $h$ values

$\bar{h}=\frac{8.5+14+20+27+10.5+17+23}{7}=\frac{120}{7}=17.142857\ldots$

2. Use the line through the mean point to find $\bar{s}$ and hence $\sum s$

The regression line of $s$ on $h$ passes through $(\bar{h},\bar{s})$, so
$\bar{s}=-3.00799(17.142857\ldots)+100.27974=48.714\ldots$
$\sum s = 7\bar{s}=7(48.714\ldots)=341.0$

3. Subtract the six known scores to obtain $\alpha$

Sum of the six known scores $=68+61+44+12+58+31=274$.
$\alpha=\sum s-274=341.0-274=67.0$

1. Set up axes with appropriate scales and labels

Draw a set of axes on graph paper. Label the horizontal axis as $h$ (number of hours of screen-time) and the vertical axis as $s$ (average score). Choose a suitable scale that covers the entire range of data (e.g., $0$ to $30$ for $h$ and $0$ to $70$ for $s$).

2. Plot the given data points

Carefully plot the given 7 pairs of values as coordinate points $(h, s)$ on the axes:

$(8.5, 68)$, $(10.5, 67)$, $(14, 61)$, $(17, 58)$, $(20, 44)$, $(23, 31)$, $(27, 12)$.

1. Evaluate and compare the product moment correlation coefficients ($r$) of both models to justify the better fit.

Using a graphing calculator to find $r$:
For the linear model $s = mh + c$ (part a), $r \approx -0.961$.
For the model $s = kh^2 + c$, we use the data points $(h^2, s)$:
$h^2$: $72.25, 110.25, 196, 289, 400, 529, 729$
$s$: $68, 67, 61, 58, 44, 31, 12$
For $s$ on $h^2$, $r \approx -0.991$.

Since the correlation coefficient for $s = kh^2 + c$ ($r \approx -0.991$) is closer to $-1$ than that of $s = mh + c$, the points $(h^2, s)$ lie closer to a straight line. Thus, $s = kh^2 + c$ is the better model.

(Note: Explaining that the scatter diagram plotted in part (b) exhibits a downward curve with a non-constant gradient, making a non-linear quadratic model more appropriate, is also an acceptable valid reason).

2. Find the values of $k$ and $c$ using linear regression on the transformed variables.

Using the graphing calculator to perform linear regression of $s$ on $h^2$:
$s = kh^2 + c$
$k \approx -0.08733$
$c \approx 77.727$

Rounding to 3 significant figures:
$k = -0.0873$
$c = 77.7$

1. Calculate the estimated score using the model from part (c)

Using the equation $s = kh^2 + c$ found in part (c), where $k \approx -0.087288$ and $c \approx 77.713$:
$s = -0.087288(7)^2 + 77.713$
$s = -0.087288(49) + 77.713$
$s \approx 73.436$
$s \approx 73.4 \text{ (to 3 s.f.)}$

2. Comment on the reliability of the estimate

The value $h = 7$ is outside the range of given data values for screen-time ($8.5 \le h \le 27$).
Since this is an extrapolation, the estimate obtained is unreliable.

1. State whether the statement is valid and provide the statistical reasoning.

The statement is not valid. Correlation does not imply causation; there could be other factors affecting the exam scores.