Question 20 - A-Level Maths

Edexcel S2 2023 October — Question 20

Exam Board	Edexcel
Module	S2 (Statistics 2)
Year	2023
Session	October
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Cumulative distribution functions
Type	Expectation from CDF/PDF
Difficulty	Moderate -0.3 This is a standard S2 question testing routine application of discrete probability distributions (finding mean from a probability distribution) and sampling distributions. Part (a) requires simple weighted average calculation, parts (b)-(c) involve listing outcomes and finding a sampling distribution which is methodical but not conceptually demanding. This is typical textbook material requiring recall and systematic application rather than problem-solving insight, making it slightly easier than average for A-level.
Spec	2.05b Hypothesis test for binomial proportion 2.05c Significance levels: one-tail and two-tail 5.01a Permutations and combinations: evaluate probabilities 5.02i Poisson distribution: random events model 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities 5.02l Poisson conditions: for modelling 5.03c Calculate mean/variance: by integration

20\% of the t -shirts made by the manufacturer are small and sell for $\pounds 10 30 \%$ of the t -shirts made by the manufacturer are medium and sell for $\pounds 12$ The rest of the t -shirts made by the manufacturer are large and sell for $\pounds 15$

Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
List all the possible combinations of the individual selling prices of these 3 t-shirts.
Find the sampling distribution of the median selling price of these 3 t-shirts.
1. A supermarket receives complaints at a mean rate of 6 per week.
2. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
3. Find the probability that, in a given week, there are
  1. fewer than 3 complaints received by the supermarket,
  2. at least 6 complaints received by the supermarket.
In a randomly selected week, the supermarket received 12 complaints.
Test, at the $5 \%$ level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
1. The continuous random variable $Y$ has cumulative distribution function given by
$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$
Find $\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)$
Find the value of $k$
Use algebraic calculus to find $\mathrm { E } ( Y )$
1. The discrete random variable $X$ is given by
$$X \sim \mathrm {~B} ( n , p )$$ The value of $n$ and the value of $p$ are such that $X$ can be approximated by a normal random variable $Y$ where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$
show that $\sigma = 6$
Hence find the value of $n$ and the value of $p$

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20\% of the t -shirts made by the manufacturer are small and sell for $\pounds 10 30 \%$ of the t -shirts made by the manufacturer are medium and sell for $\pounds 12$

The rest of the t -shirts made by the manufacturer are large and sell for $\pounds 15$\\
(a) Find the mean value of the t -shirts made by the manufacturer.

A random sample of 3 t -shirts made by the manufacturer is taken.\\
(b) List all the possible combinations of the individual selling prices of these 3 t-shirts.\\
(c) Find the sampling distribution of the median selling price of these 3 t-shirts.

\begin{enumerate}
  \item A supermarket receives complaints at a mean rate of 6 per week.\\
(a) State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.\\
(b) Find the probability that, in a given week, there are\\
(i) fewer than 3 complaints received by the supermarket,\\
(ii) at least 6 complaints received by the supermarket.
\end{enumerate}

In a randomly selected week, the supermarket received 12 complaints.\\
(c) Test, at the $5 \%$ level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.\\
State your hypotheses clearly.

Following changes made by the supermarket, it received 26 complaints over a 6-week period.\\
(d) Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.

\begin{enumerate}
  \item The continuous random variable $Y$ has cumulative distribution function given by
\end{enumerate}

$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 
0 & y < 0 \\
\frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\
\frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\
1 & y > 6
\end{array} \right.$$

(a) Find $\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)$\\
(b) Find the value of $k$\\
(c) Use algebraic calculus to find $\mathrm { E } ( Y )$

\begin{enumerate}
  \item The discrete random variable $X$ is given by
\end{enumerate}

$$X \sim \mathrm {~B} ( n , p )$$

The value of $n$ and the value of $p$ are such that $X$ can be approximated by a normal random variable $Y$ where

$$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$

Given that when using a normal approximation

$$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$

(a) show that $\sigma = 6$\\
(b) Hence find the value of $n$ and the value of $p$

\hfill \mbox{\textit{Edexcel S2 2023 Q20}}

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