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IAL 2018 Oct S2 Q1

A Level / Edexcel / S2

IAL 2018 Oct Paper · Question 1

题目

Problem

Each day a restaurant opens between 11 am and 11 pm. During its opening hours, the restaurant receives calls for reservations at an average rate of 6 per hour.

(a) Find the probability that the restaurant receives exactly 1 call for a reservation between 6 pm and 7 pm.

(2)

The restaurant distributes leaflets to local residents to try and increase the number of calls for reservations. After distributing the leaflets, it records the number of calls for reservations it receives over a 90 minute period.

Given that it receives 14 calls for reservations during the 90 minute period,

(b) test, at the 5% level of significance, whether the rate of calls for reservations has increased from 6 per hour. State your hypotheses clearly.

(5)
题目中文翻译

每天一家餐厅从上午 11 点营业到晚上 11 点。在营业时间内，该餐厅接到预订电话的平均速率为每小时 6 个。

(a) 求餐厅在下午 6 点到 7 点之间恰好接到 1 个预订电话的概率。

(b) 餐厅向当地居民分发传单，试图增加预订电话的数量。发放传单后，它记录了在 90 分钟内接到的预订电话数量。

已知在这 90 分钟内接到 14 个预订电话。

在 5% 显著性水平下检验：预订电话的到达速率是否从每小时 6 个增加了。清楚写出假设。

解答

(a)

解法一

思路

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一小时内收到的预订电话数服从均值为 6 的泊松分布。把电话数取 1，代入泊松分布的单点概率公式。

答题过程

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Let XX be the number of calls received between 6 pm and 7 pm. Then

X∼Po⁡(6).X\sim\operatorname{Po}(6).

Therefore,

P(X=1)= e−6611!= 6e−6= 0.0148725…≈ 0.0149.\begin{align*} P(X=1) =&\,\frac{e^{-6}6^1}{1!} \\ =&\,6e^{-6} \\ =&\,0.0148725\ldots \\ \approx&\,\boxed{0.0149}. \end{align*}

(b)

解法一

思路

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设 λ\lambda 为每小时预订电话的平均到达率。因为要检验到达率是否增加，使用上尾检验；在原假设下，90 分钟内的期望电话数为 6×1.5=96\times1.5=9。计算观测到至少 14 个电话的上尾概率，并与 0.050.05 比较。

答题过程

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Let λ\lambda be the mean number of reservation calls per hour. The hypotheses are

H0:λ=6,H1:λ>6.H_0:\lambda=6, \qquad H_1:\lambda>6.

Let YY be the number of calls received in 90 minutes. Under H0H_0,

Y∼Po⁡(6×1.5)=Po⁡(9).Y\sim\operatorname{Po}(6\times1.5) =\operatorname{Po}(9).

The pp-value is

P(Y⩾14)= 1−P(Y⩽13)= 1−0.9261= 0.0739.\begin{align*} P(Y\geqslant14) =&\,1-P(Y\leqslant13) \\ =&\,1-0.9261 \\ =&\,0.0739. \end{align*}

Since 0.0739>0.050.0739>0.05, do not reject H0H_0. There is insufficient evidence at the 5% significance level to suggest that the rate of reservation calls has increased.

解法二

思路

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也可先求 5% 显著性水平下的上尾临界域。找到最小的临界电话数，使落入该上尾区域的概率不超过 0.050.05，再判断观测值 14 是否落入临界域。

答题过程

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Using the same hypotheses and Y∼Po⁡(9)Y\sim\operatorname{Po}(9) under H0H_0,

P(Y⩾15)=0.0415<0.05.P(Y\geqslant15)=0.0415<0.05.

Also, from the first method,

P(Y⩾14)=0.0739>0.05.P(Y\geqslant14)=0.0739>0.05.

Therefore, the critical region is

Y⩾15.Y\geqslant15.

The observed value, Y=14Y=14, is not in the critical region, so do not reject H0H_0. There is insufficient evidence at the 5% significance level to suggest that the rate of reservation calls has increased.