Statistics 1
Pearson Edexcel · International A-Level · Mathematics トピック 6 5:20 英語ナレーション・英語＋中文字幕 burning-in
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Transcript
Paper WST01: 1 hour 30 minutes, 75 marks, with P1 and P2 assumed.
WST01 试卷：1 小时 30 分钟，75 分，默认已学 P1 和 P2。
The booklet carries the tables; these formulas are yours: means, the addition and multiplication rules, coding for expectation and variance, and standardising a Normal variable.
公式册里有统计表；这些公式要自己记：平均数、加法和乘法法则、期望和方差的编码，以及正态变量的标准化。
A statistical model is a simplified description of a real situation.
统计模型是对真实情境的简化描述。
It is quicker and cheaper than the real experiment, it can be repeated and refined, and it exposes relationships.
它比真实实验更快、更便宜，可以重复和改进，并能揭示关系。
Give more than one reason.
要给出不止一个理由。
Speeds of 225 cars in unequal classes have densities 10.8, 18, 6, 1 and 0.4.
225 辆车的速度分在不等宽的组里，频率密度为 10.8、18、6、1 和 0.4。
The tallest bar is drawn 10 centimetres tall, so the second-tallest is 10.8 over 18 of that: 6 centimetres.
最高的柱画成 10 厘米，所以第二高的是它的 10.8/18：6 厘米。
Beware: the 30 to 40 bar ALSO has density 6; only the ratio earns the marks.
注意：30 到 40 的柱密度也是 6；只有比例推理才得分。
Class A has 24 students averaging 47; class B, 30 students; together, 54 averaging 45.
A 班 24 人，平均 47；B 班 30 人；合起来 54 人，平均 45。
The total is 2430, so B's total is 1302 and b is 43.4.
总和是 2430，所以 B 班总和是 1302，b = 43.4。
The sums of squares add, giving a combined standard deviation of 24.1.
平方和直接相加，合并标准差是 24.1。
Raise every B mark by 2 and B moves closer to A: the combined spread falls.
把 B 班每个分数加 2，B 更接近 A：合并后的离散程度下降。
The median is the 112.5th value, inside the 25 to 30 class: 25 plus 5 times 58.5 over 90, which is 28.25.
中位数是第 112.5 个值，在 25 到 30 组内：25 + 5 × 58.5/90，即 28.25。
The upper quartile is the 168.75th, inside 30 to 40: 34.125.
上四分位数是第 168.75 个，在 30 到 40 组内：34.125。
With the lower quartile 25.14, the interquartile range is 8.985.
下四分位数是 25.14，所以四分位距是 8.985。
Compare the two halves of the box.
比较箱子的两半。
From the median to the upper quartile is 5.875; from the lower quartile to the median, only 3.11.
从中位数到上四分位数是 5.875；从下四分位数到中位数只有 3.11。
The mean, 30.6, is above the median too: the speeds are positively skewed.
平均数 30.6 也高于中位数：速度呈正偏态。
The overlap is P of C given D times P of D: 0.2 times 0.45, which is 0.09.
交集等于 P(C|D) × P(D)：0.2 × 0.45 = 0.09。
Then the union rule gives P of C is 0.23, while P of D is 0.45.
再由并集公式得 P(C) = 0.23，而 P(D) = 0.45。
Fill the diagram and both answers fall out.
填好图，两个答案都出来了。
The first draw is yellow with 6 over 9, white with 3 over 9.
第一次抽到黄色的概率是 6/9，白色是 3/9。
Without replacement, the second layer drops by one: after yellow, 5 over 8 and 3 over 8; after white, 6 over 8 and 2 over 8.
不放回，第二层分子分母各减一：黄色之后是 5/8 和 3/8；白色之后是 6/8 和 2/8。
Exactly one yellow is a half.
恰好一个黄色的概率是 1/2。
Given at least one white, what is the chance both are white?
已知至少一个白色，两个都是白色的概率是多少？
The condition, at least one white, has probability 7 over 12, and both white has 1 over 12.
条件“至少一个白色”的概率是 7/12，两个都是白色是 1/12。
Divide: one seventh.
相除：1/7。
Gas use is coded as w equals 11.5 d.
用气量编码为 w = 11.5d。
S w w is 445.57, so r is minus 0.844: warmer days, less gas.
S_ww = 445.57，所以 r = −0.844：天越暖，用气越少。
The regression of w on t has gradient minus 3.46 and intercept 30.9; dividing by 11.5 gives d on t.
w 对 t 的回归直线斜率 −3.46，截距 30.9；除以 11.5 得 d 对 t 的直线。
The regression line of y on x predicts y from x, and only within the data range.
y 对 x 的回归直线只能由 x 预测 y，而且只能在数据范围内。
Swap the variables and the line is different, and extrapolation is the flagged danger.
交换变量直线就不同了，外推是被特别警告的危险。
Solve the inequality for X first: 3X minus 3 less than X plus 2 means X is below 2.5, probability 0.5.
先解关于 X 的不等式：3X − 3 < X + 2 即 X < 2.5，概率 0.5。
Then E of X is 2.35 and E of X squared 6.65, so the variance is 1.1275, and coding multiplies it by 4: 4.51.
然后 E(X) = 2.35，E(X²) = 6.65，所以方差是 1.1275，编码后乘以 4：4.51。
Small bags of flour are N of 502, 3 squared.
小袋面粉服从 N(502, 3²)。
508 is two standard deviations above the mean, so the chance of a bag above 508 is the tail beyond Z equals 2: 0.0228.
508 比平均数高两个标准差，所以一袋超过 508 的概率就是 Z = 2 以外的尾部：0.0228。
The 85th percentile gives z of 1.0364, so sigma is 14.
第 85 百分位数给出 z = 1.0364，所以 σ = 14。
If P of X above k equals P of Y below 2k, the two z values match: k minus 502 over 3 equals 1024 minus 2k over 14.
若 P(X > k) = P(Y < 2k)，两个 z 值相等：(k − 502)/3 = (1024 − 2k)/14。
So k is 505, and the probability is P of Z above 1: 0.159.
所以 k = 505，概率是 P(Z > 1) = 0.159。
Check yourself.
自我检测。
A histogram bar's height is frequency density.
直方图柱高是频率密度。
A longer upper half means positive skew.
上半部分更长表示正偏态。
P of B given A is P of A and B over P of A.
P(B|A) = P(A∩B)/P(A)。
Var of aX plus b is a squared Var X.
Var(aX + b) = a²Var(X)。
And a regression of y on x predicts y from x, within range.
y 对 x 的回归在范围内由 x 预测 y。