- The random variable \(A\) is defined as
$$A = 4 X - 3 Y$$
where \(X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Find
- \(\mathrm { E } ( A )\),
- \(\operatorname { Var } ( A )\).
The random variables \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\) and \(Y _ { 4 }\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as
$$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
- Find \(\mathrm { P } ( B > A )\).
Paper Reference(s)
\section*{6691/01 Edexcel GCE}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Examiner's use only}
\includegraphics[alt={},max width=\textwidth]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-041_97_306_495_1635}
\end{figure}
\(0 - 3\) & 8
\hline
\(3 - 5\) & 12
\hline
\(5 - 6\) & 13
\hline
\(6 - 8\) & 9
\hline
\(8 - 12\) & 8
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table} - Show that an estimate of \(\bar { X } = 5.49\) and an estimate of \(S _ { X } ^ { 2 } = 6.88\)
The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2.
\begin{table}[h]
| Waiting Time | \(\mathrm { x } < 3\) | \(3 - 5\) | \(5 - 6\) | \(6 - 8\) | \(\mathrm { x } > 8\) |
| Expected Frequency | 8.56 | 12.73 | 7.56 | a | b |
\captionsetup{labelformat=empty}
\caption{Table 2}
- Find the value of a and the value of b .
- Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.
\section*{Q uestion 4 continued}
- Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
One large and 3 small bottles of Blumen are chosen at random. - Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles.
A large bottle and a small bottle of Blumen are chosen at random.
- Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
\section*{Q uestion 5 continued}
6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg . - Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
- Explain the relevance of the Central Limit Theorem to the test in part (a).
\section*{Q uestion 6 continued}
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
7. Lambs are born in a shed on M ill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below.
$$\begin{array} { l l l l l l l l }
4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40
\end{array}$$ - Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm.
A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
- Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean.
The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
- Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
\section*{Q uestion 7 continued}