Standard +0.3 This is a standard normal distribution question covering routine techniques: z-score calculations, inverse normal for percentiles, conditional probability with truncated distributions, and a straightforward hypothesis test. Part (e-f) add basic probability calculations. All parts follow textbook methods with no novel insight required, making it slightly easier than average.
Jake works for a parcel delivery company. The masses, in kilograms, of parcels he delivers are normally distributed with mean \(2 \cdot 2\) and standard deviation \(0 \cdot 3\).
Calculate the probability that a randomly selected parcel will have a mass less than 1.8 kg .
Jake delivers the lightest \(80 \%\) of parcels on his bike. The rest he puts in his car and delivers by car.
Find the mass of the heaviest parcel he would deliver by bike.
He randomly selects a parcel from his car. Find the probability that it has a mass less than 3 kg .
In the run-up to Christmas, Jake believes that the parcels he has to deliver are, on average, heavier. He assumes that the standard deviation is unchanged. He randomly selects 20 parcels and finds that their total mass is 46 kg . Test Jake's belief at the \(5 \%\) level of significance.
Jake delivers each parcel to one of three areas, \(A , B\) or \(C\). The probabilities that a parcel has destination area \(A , B\) and \(C\) are \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. All parcels are considered to be independent.
On a particular day, Jake has three parcels to deliver. Find the probability that he will have to deliver to all three areas.
On a different day, Jake has two parcels to deliver. Find the probability that he will have to deliver to more than one area.
\begin{enumerate}
\item Jake works for a parcel delivery company. The masses, in kilograms, of parcels he delivers are normally distributed with mean $2 \cdot 2$ and standard deviation $0 \cdot 3$.\\
(a) Calculate the probability that a randomly selected parcel will have a mass less than 1.8 kg .\\
\end{enumerate}
Jake delivers the lightest $80 \%$ of parcels on his bike. The rest he puts in his car and delivers by car.\\
(b) Find the mass of the heaviest parcel he would deliver by bike.\\
(c) He randomly selects a parcel from his car. Find the probability that it has a mass less than 3 kg .\\
(d) In the run-up to Christmas, Jake believes that the parcels he has to deliver are, on average, heavier. He assumes that the standard deviation is unchanged. He randomly selects 20 parcels and finds that their total mass is 46 kg . Test Jake's belief at the $5 \%$ level of significance.
Jake delivers each parcel to one of three areas, $A , B$ or $C$. The probabilities that a parcel has destination area $A , B$ and $C$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$ respectively. All parcels are considered to be independent.\\
(e) On a particular day, Jake has three parcels to deliver. Find the probability that he will have to deliver to all three areas.\\
(f) On a different day, Jake has two parcels to deliver. Find the probability that he will have to deliver to more than one area.\\
\hfill \mbox{\textit{WJEC Unit 4 2024 Q4 [21]}}