人们认为喷药消杀的次数与新增登革热病例的数量之间存在相关性。为了测试喷药消杀对控制登革热传播的效果，在一个月中定期对特定社区进行若干次喷药消杀（ $x$ ），并记录相应的新增登革热病例数（ $y$ ）。结果汇总在下表中：

((i))
计算 $x$ 与 $y$ 之间的积差相关系数，并解释你的答案是否表明线性模型是合适的。
((ii))
绘制该数据的散点图。
((iii))
其中一个数据点似乎是错误的。

通过将其标记为 $P$ 来指出该点，并解释为什么其余点的散点图可能与形式为

$y = Ae^{bx}$

的模型一致。
((iv))
省略 $P$ ，使用适当的回归线给出当新增登革热病例数为 7 时，一个月内喷药消杀次数的最佳估计值（四舍五入到最接近的整数）。

它不会直接告诉你答案，而是问你卡在哪一步看看 AI 是怎么陪你一步步弄懂这道题的

完整解法

8((i))

1. Calculate the product moment correlation coefficient $r$ using a calculator

Entering the values of $x$ and $y$ into a graphing calculator:
$r = -0.95509...$
$r = -0.955$ (to 3 s.f.)

2. Explain whether the value of $r$ suggests a linear model is appropriate

Since $r = -0.955$ is close to $-1$ , it indicates a strong negative linear correlation between $x$ and $y$ . Therefore, based on this value alone, it suggests that a linear model is appropriate.

答案

$r = -0.955$
Since $r$ is close to $-1$ , it suggests a strong negative linear correlation, meaning a linear model is appropriate.

8((ii))

1. Draw axes, determine scales, and plot the given data points

Set up a pair of axes with appropriate linear scales to accommodate the domains $0 \le x \le 14$ and $0 \le y \le 20$ .
Plot the given points accurately: $(2, 20)$ , $(4, 14)$ , $(6, 9)$ , $(8, 6)$ , $(10, 7)$ , $(12, 2)$ , and $(14, 1)$ .
Label the axes clearly as 'No. of fogging sessions, $x$ ' and 'No. of new dengue cases, $y$ '.

答案

[A properly scaled scatter diagram with the points $(2, 20)$ , $(4, 14)$ , $(6, 9)$ , $(8, 6)$ , $(10, 7)$ , $(12, 2)$ , and $(14, 1)$ plotted, with the horizontal axis labelled $x$ and the vertical axis labelled $y$ .]

8((iii))

1. Identify the incorrect data point $P$ from the anomaly in the trend.

Observing the general trend, as $x$ increases, $y$ is decreasing. However, from $x=8$ to $x=10$ , $y$ incorrectly increases from $6$ to $7$ . Thus, the anomalous point is $(10, 7)$ .

$P(10, 7)$

2. Explain why the remaining points are consistent with an exponential model.

Excluding $P$ , the remaining points on the scatter diagram form a curve where the gradient is negative but increasing towards $0$ (i.e. becoming less steep). This means $y$ decreases at a decreasing rate as $x$ increases, which visually matches the shape of an exponential curve $y = Ae^{bx}$ with $A > 0$ and $b < 0$ .

答案

$P$ is the point $(10, 7)$ .

The scatter diagram of the remaining points shows that as $x$ increases, $y$ decreases at a decreasing rate. This non-linear, concave upward decreasing trend is characteristic of an exponential decay curve $y = Ae^{bx}$ where $A > 0$ and $b < 0$ .

8((iv))

1. Identify the appropriate regression line and find its equation using the valid data points.

Since the number of fogging sessions ( $x$ ) is predetermined and spread out regularly, $x$ is the independent (controlled) variable. Therefore, the appropriate regression line to use is the regression line of $\ln y$ on $x$ .

Omitting the anomalous point $P(10, 7)$ , we use the remaining 6 data points to obtain the equation from a graphing calculator:
$\ln y = -0.24888x + 3.6276$

2. Substitute the given value of $y$ to estimate $x$.

Substitute $y = 7$ into the regression equation to find the corresponding value of $x$ :
$\ln 7 = -0.24888x + 3.6276$
$0.24888x = 3.6276 - 1.9459$
$0.24888x = 1.6817$
$x \approx 6.757$

Rounding to the nearest whole number, the best estimate for the number of fogging sessions is $7$ .

答案

$7$

AI 分析，可能出错

用 AI 批改，圈出写错的那一步，并自动将你的薄弱点存入错题本，形成你的错因库。

完整解法

8((i))

8((ii))

8((iii))

8((iv))

同一知识点的其他题