Moderate -0.3 This is a straightforward application of standard results for linear combinations of normal random variables. Parts (a) and (b) require direct use of formulas for E(aX + bY) and Var(aX + bY), while part (c) involves finding P(B > A) by recognizing B - A is normal and standardizing. All steps are routine S3 material with no novel insight required, making it slightly easier than average.
$$A = 4 X - 3 Y$$
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 \(\mathrm { P } ( B > A )\).
advancing learning, changing lives
A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.
Test, at the \(5 \%\) level of significance, the managing director's claim. State your hypotheses clearly.
2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.
Find the probability that James' time for the qualifying lap is less than Philip's.
The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
find the probability that Philip beats James in the race by more than 2 minutes.
3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .
Find the probability that \(X\) is within 0.6 mm of \(w\).
The same board is measured 16 times and the results are recorded.
Find the probability that the mean of these results is within 0.3 mm of \(w\).
Given that the mean of these 16 measurements is 35.6 mm ,
find a \(98 \%\) confidence interval for \(w\).
A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.
advancing learning, changing lives
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-055_2632_1828_123_121}
2. A county councillor is investigating the level of hardship, h , of a town and the number of calls per 100 people to the emergency services, c. He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-081_2642_1833_118_118}
2. A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than \(5 \%\).
(5)
3. The table below shows the population and the number of council employees for different towns and villages.
\end{table}
A nswers without working may not gain full credit.
\begin{figure}[h]
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]
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}
\end{figure}
\begin{enumerate}
\item The random variable $A$ is defined as
\end{enumerate}
$$A = 4 X - 3 Y$$
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\\
(a) $\mathrm { E } ( A )$,\\
(b) $\operatorname { Var } ( A )$.
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
$$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
(c) Find $\mathrm { P } ( B > A )$.\\
advancing learning, changing lives
\begin{enumerate}
\item A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.
\end{enumerate}
Test, at the $5 \%$ level of significance, the managing director's claim. State your hypotheses clearly.\\
2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.\\
(a) Find the probability that James' time for the qualifying lap is less than Philip's.
The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,\\
(b) find the probability that Philip beats James in the race by more than 2 minutes.\\
3. A woodwork teacher measures the width, $w \mathrm {~mm}$, of a board. The measured width, $X \mathrm {~mm}$, is normally distributed with mean $w \mathrm {~mm}$ and standard deviation 0.5 mm .\\
(a) Find the probability that $X$ is within 0.6 mm of $w$.
The same board is measured 16 times and the results are recorded.\\
(b) Find the probability that the mean of these results is within 0.3 mm of $w$.
Given that the mean of these 16 measurements is 35.6 mm ,\\
(c) find a $98 \%$ confidence interval for $w$.\\
\begin{enumerate}
\item A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, $b \mathrm {~cm}$, and the depth of a river, $s \mathrm {~cm}$, at seven positions. The results are shown in the table below.
\end{enumerate}
advancing learning, changing lives\\
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-055_2632_1828_123_121}\\
2. A county councillor is investigating the level of hardship, h , of a town and the number of calls per 100 people to the emergency services, c. He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
\begin{enumerate}
\item Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
\end{enumerate}
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-081_2642_1833_118_118}\\
2. A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than $5 \%$.\\
(5)\\
3. The table below shows the population and the number of council employees for different towns and villages.
\end{table}
A nswers without working may not gain full credit.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{
$0 - 3$ & 8 \\
\hline
$3 - 5$ & 12 \\
\hline
$5 - 6$ & 13 \\
\hline
$6 - 8$ & 9 \\
\hline
$8 - 12$ & 8 \\
\hline
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
(a) 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]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Waiting Time & $\mathrm { x } < 3$ & $3 - 5$ & $5 - 6$ & $6 - 8$ & $\mathrm { x } > 8$ \\
\hline
Expected Frequency & 8.56 & 12.73 & 7.56 & a & b \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{center}
\end{table}
(b) Find the value of a and the value of b .\\
(c) Test, at the $5 \%$ level of significance, the manager's belief. State your hypotheses clearly.\\
\section*{Q uestion 4 continued}
\begin{enumerate}
\item 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 .
\end{enumerate}
One large and 3 small bottles of Blumen are chosen at random.\\
(a) 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.\\
(b) 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 .\\
(a) 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.\\
(b) 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.
$$\begin{array} { l l l l l l l l }
4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40
\end{array}$$
(a) 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.\\
(b) 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 .\\
(c) 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}
\end{figure}
\hfill \mbox{\textit{Edexcel S3 Q8}}