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Date	May Specimen paper	Marks available	6	Reference code	SPM.1.AHL.TZ0.10
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find	Question number	10	Adapted from	N/A

Question

In a coffee shop, the time it takes to serve a customer can be modelled by a normal distribution with a mean of 1.5 minutes and a standard deviation of 0.4 minutes.

Two customers enter the shop together. They are served one at a time.

Find the probability that the total time taken to serve both customers will be less than 4 minutes.

Clearly state any assumptions you have made.

Markscheme

let $T$ $T$ be the time to serve both customers and $T_{i}$ ${T_i}$ the time to serve the $i$ $i$ th customer

assuming independence of $T_{1}$ ${T_1}$ and $T_{2}$ ${T_2}$ R1

$T$ $T$ is normally distributed and $T = T_{1} + T_{2}$ $T = {T_1} + {T_2}$ (M1)

$E (T) = 1.5 + 1.5 = 3$ $E\left( T \right) = 1.5 + 1.5 = 3$ A1

$Var (T) = {0.4}^{2} + {0.4}^{2} = 0.32$ ${\text{Var}}\left( T \right) = {0.4^2} + {0.4^2} = 0.32$ M1A1

$P (T < 4) = 0.961$ $P\left( {T < 4} \right) = 0.961$ A1

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 4—Statistics and probability » AHL 4.15—Central limit theorem

19M.3.AHL.TZ0.Hsp_3e.ii:
Write down the conclusion reached.
22M.1.AHL.TZ1.9b:
A statistician in the company suggests it would be fairer if the company passes the inspection when the mean weight of five randomly chosen bags is greater than $950 g$ $950 g$ .

Find the probability of passing the inspection if the statistician’s suggestion is followed.
21M.1.AHL.TZ1.14a:
Find the mean weight of a bag of apples.
21N.1.AHL.TZ0.14a:
Find the probability that a sack is under its labelled weight.
21N.1.AHL.TZ0.14b:
Find the lower quartile of the weights of the sacks of potatoes.
16N.3.AHL.TZ0.Hsp_2b:
Explain why a normal distribution can be used to give an approximate model for $X$ $X$ .
19M.3.AHL.TZ0.Hsp_3d:
Find a 90 % confidence interval for $μ$ $\mu$ .
16N.3.AHL.TZ0.Hsp_2d:
Calculate the probability that he makes a Type II error.
19M.3.AHL.TZ0.Hsp_3c:
Find an unbiased estimate for $σ^{2}$ ${\sigma ^2}$ .
19M.3.AHL.TZ0.Hsp_3a:
State the distribution of $¯ X$ $\bar X$ giving its mean and variance.
16N.3.AHL.TZ0.Hsp_2a:
Find

(i) $E (X)$ ${\text{E}}(X)$ ;

(ii) $Var (X)$ ${\text{Var}}(X)$ .
19N.3.AHL.TZ0.Hsp_3a:
State the central limit theorem as applied to a random sample of size $n$ $n$ , taken from a distribution with mean $μ$ $\mu$ and variance $σ^{2}$ ${\sigma ^2}$ .
19M.3.AHL.TZ0.Hsp_3e.i:
Find the $p$ $p$ -value for the test.
19M.3.AHL.TZ0.Hsp_3b:
Find an unbiased estimate for $μ$ $\mu$ .
16N.3.AHL.TZ0.Hsp_2c:
Use this model to find the values of $A$ $A$ and $B$ $B$ such that $P (A < X < B) = 0.9$ ${\text{P}}(A < X < B) = 0.9$ , where $A$ $A$ and $B$ $B$ are symmetrical about the mean of $X$ $X$ .
21M.1.AHL.TZ1.14c:
Find the probability that a bag selected at random weighs more than $1 kg$ $1 kg$ .
21M.1.AHL.TZ1.14b:
Find the standard deviation of the weights of these bags of apples.
21N.1.AHL.TZ0.14c:
The sacks of potatoes are transported in crates. There are $10$ $10$ sacks in each crate and the weights of the sacks of potatoes are independent of each other.

Find the probability that the total weight of the sacks of potatoes in a crate exceeds $500 kg$ $500 kg$ .

Topic 4—Statistics and probability

Question

Markscheme

Examiners report

Syllabus sections

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