Pearson Edexcel International Advanced Level

The number of cars passing a motorway checkpoint can be modelled by a Poisson distribution with mean 3.6 per minute.

(a) Find the probability that exactly 4 cars pass the checkpoint in a randomly chosen 1-minute period. (2)

(b) Find the probability that at least 3 cars pass the checkpoint in a randomly chosen 1-minute period. (2)

(c) Use a suitable approximation to estimate the probability that more than 20 cars pass the checkpoint in a randomly chosen 5-minute period.
(Solutions relying entirely on calculator technology are not acceptable.) (3)

Given that 12 cars pass the checkpoint in a randomly chosen 4-minute period,

(d) find the probability that exactly 5 of these 12 cars pass in the first 2 minutes of the 4-minute period. (1)

The continuous random variable $X$ has probability density function

$$\mathrm{f}(x) = \begin{cases} kx(4-x), & 0 \leqslant x \leqslant 2, \\ 0, & \text{otherwise.} \end{cases}$$

(a) Show that $k = \dfrac{3}{16}$. (3)

(b) Find $\mathrm{P}(0.5 < X < 1.5)$. (2)

(c) Find $\mathrm{E}(X)$. (2)

(d) Explain why the mode of $X$ is 2.
(You may assume that $\mathrm{f}(x)$ has a single stationary point.) (2)

(e) Find $\mathrm{P}\bigl(X > \mathrm{E}(X)\bigr)$. (2)

(f) Show that the median of $X$ lies between 1.30 and 1.31. (1)

(g) Sketch the probability density function of $X$. (1)

A company produces electronic components. The company claims that the proportion of defective components is 8%.

A quality control inspector randomly selects 30 components and finds the number, $X$, that are defective.

(a) State the distribution of $X$. (1)

(b) Find the probability that exactly 2 of the 30 components are defective. (2)

(c) Find the probability that at least 4 of the 30 components are defective. (2)

The inspector finds that 6 of the 30 components are defective. The inspector suspects the true proportion of defective components may be greater than 8%.

(d) Test, at the 5% level of significance, whether the inspector’s suspicion is justified. State your hypotheses clearly. (4)

(e) State the critical region for this test. (2)

A hotel manager knows from past records that the number of complaints received per week can be modelled by a Poisson distribution with mean 3.

Following staff training, the manager records the number of complaints received over a 4-week period and finds a total of 6 complaints.

(a) Test, at the 5% level of significance, whether the mean number of complaints per week has decreased. State your hypotheses clearly. (7)

(b) State the critical region for this test. (1)

(c) State the condition required for the Poisson distribution to be a suitable model in this context. (2)

(d) Explain whether your conclusion would be different if the test were carried out at the 1% level of significance. (2)

The number of customers arriving at a supermarket checkout per hour can be modelled by a Poisson distribution with mean 8.4.

(a) Find the probability that exactly 6 customers arrive in a randomly chosen 1-hour period. (2)

(b) Find the probability that at most 5 customers arrive in a randomly chosen 1-hour period. (2)

(c) Use a suitable approximation to estimate the probability that more than 55 customers arrive in a randomly chosen 6-hour period.
(Solutions relying entirely on calculator technology are not acceptable.) (5)

Given that 24 customers arrive in a 3-hour period,

(d) find the probability that exactly 10 of these 24 customers arrive in the first hour of the 3-hour period. (2)

The continuous random variable $X$ has probability density function

$$\mathrm{f}(x) = \begin{cases} \dfrac{3}{64}(x-2)^2(6-x), & 2 \leqslant x \leqslant 6, \\ 0, & \text{otherwise.} \end{cases}$$

(a) Show that $\mathrm{f}(x)$ is a valid probability density function. (2)

(b) Find $\mathrm{E}(X)$. (2)

(c) Find $\operatorname{Var}(X)$. (2)

(d) Find the mode of $X$. (2)

(e) Without further calculation, state with a reason whether the distribution of $X$ is positively or negatively skewed. (2)

(f) Show that the median of $X$ lies between 4.45 and 4.46. (1)

In a large population, it is known that 28% of people have blood type O+.

A random sample of 50 people is selected from this population.

(a) Find the probability that between 10 and 20 inclusive of these 50 people have blood type O+. (2)

A random sample of 180 people is selected from this population.

(b) Use a normal approximation to estimate the probability that between 40 and 60 inclusive of these 180 people have blood type O+. You must apply a continuity correction. (3)

(c) Explain why a continuity correction is needed in this context. (1)

(d) State the conditions under which a normal approximation to a binomial distribution is valid. (1)

The number of calls received by a call centre in an hour can be modelled by a Poisson distribution with mean 40.

(e) Use a normal approximation to estimate the probability that at most 35 calls are received in a randomly chosen hour. You must apply a continuity correction. (2)