(12 points)
To monitor the production process of a production line for a certain component, an inspector randomly selects one component every 30 minutes and measures its size (in cm). Below are the sizes of 16 components randomly selected by the inspector in one day:

Sampling Order	1	2	3	4	5	6	7	8
Component Size	9.95	10.12	9.96	9.96	10.01	9.92	9.98	10.04
Sampling Order	9	10	11	12	13	14	15	16
Component Size	10.26	9.91	10.13	10.02	9.22	10.04	10.05	9.95

$\sqrt{\sum_{i=1}^{16}(i - 8.5)^2} \approx 18.439$, $\sum_{i=1}^{16}(x_i - \bar{x})(i - 8.5) = -2.78$, where $x_i$ is the size of the $i$-th component sampled, $i = 1, 2, \cdots, 16$.
(1) Find the correlation coefficient $r$ of $(x_i, i)$ $(i = 1, 2, \cdots, 16)$, and determine whether it can be concluded that the size of components produced on this day does not systematically increase or decrease as the production process progresses.
(2) Among the components sampled in one day, if a component with size outside $(\bar{x} - 3s, \bar{x} + 3s)$ appears, it is considered that the production line may have experienced an abnormal situation on this day, and the production process needs to be checked.
(i) Based on the sampling results of this day, is it necessary to check the production process?
(ii) Data outside $(\bar{x} - 3s, \bar{x} + 3s)$ are called outliers. Remove the outliers and estimate the mean and standard deviation of the component sizes produced by this production line on this day. (Round to 0.01)
$$\text{Attachment: For a sample }(x_i, y_i) (i = 1, 2, \cdots, n), \text{ the correlation coefficient is } r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}.$$
$$\sqrt{0.008} \approx 0.09.$$

Based on your answers to the previous questions, give your opinion on the speed of convergence of the $d_{ii}^{(m)}$ to the eigenvalues of $\Sigma$.

To understand whether IQ and brain volume are related, a small study used magnetic resonance imaging to measure the brain volume (in units of 10,000 pixels) of 5 people, along with their IQ listed in the table below:

Brain Volume $( X )$	90	95	91	88	106
$\mathrm { IQ } ( Y )$	90	100	112	80	103

It is known that the mean of $X$ in the table above is $\mu _ { X } = 94$ , the mean of $Y$ is $\mu _ { Y } = 97$ , and the correlation coefficient between brain volume ($X$) and IQ ($Y$) is $r _ { X , Y }$ . Based on the table above, determine which of the following options is most likely the value of $r _ { X , Y }$ ?
(1) $r _ { X , Y } \leq - 1$
(2) $- 1 < r _ { X , Y } < - 0.5$
(3) $r _ { X , Y } = 0$
(4) $0 < r _ { X , Y } < 0.5$
(5) $r _ { X , Y } \geq 1$