It is estimated that \(4 \%\) of people have green eyes. In a random sample of size \(n\), the expected number of people with green eyes is 5 .
Calculate the value of \(n\).
The expected number of people with green eyes in a second random sample is 3 .
Find the standard deviation of the number of people with green eyes in this second sample. expected number of people with green eyes is 5 .
Calculate the value of \(n\) -
The expected number of people with green eyes in a second random sample is 3 .
sample. C)
It is estimated that \(4 \%\) of people have green eyes. In a random sample of size \(n\), the
expected number of people with green eyes is 5 .
Calculate the value of \(n\).
The expected number of people with green eyes in a second random sample is 3 .
Find the standard deviation of the number of people with green eyes in this second
sample.
The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).
Write down the probability density function \(\mathrm { f } ( x )\).
Find
\(\mathrm { E } ( X )\),
\(\operatorname { Var } ( X )\),
the cumulative distribution function of \(X\), for all \(x\),
\(\mathrm { P } ( 2.3 < X < 3.4 )\).
3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.
State two conditions under which a Poisson distribution is a suitable model for \(X\).
The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
a randomly chosen page has no misprints,
the total number of misprints on 2 randomly chosen pages is more than 7 .
The first chapter contains 20 pages.
Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
4. Explain what you understand by
a sampling unit,
a sampling frame,
a sampling distribution.
5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.
Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
Estimate the probability there are less than 5 defective articles.
6. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where
$$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2 \\ 0 , & \text { otherwise } \end{cases}$$
where \(k\) is a positive integer.
Show that \(k = \frac { 1 } { 4 }\).
Find
\(\mathrm { E } ( X )\),
the mode of \(X\),
the median of \(X\).
Comment on the skewness of the distribution.
Sketch f(x).
7. A drugs company claims that \(75 \%\) of patients suffering from depression recover when treated with a new drug.
A random sample of 10 patients with depression is taken from a doctor's records.
Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug.
Given that the claim is correct,
find the probability that the treatment will be successful for exactly 6 patients.
The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the \(1 \%\) level.
Turn over
Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.
Explain why the secretary decided to take a random sample of club members rather than ask all the members.
The continuous random variable \(L\) represents the error, in mm , made when a machine cuts rods to a target length. The distribution of \(L\) is continuous uniform over the interval [-4.0, 4.0].
Find
\(\mathrm { P } ( L < - 2.6 )\),
\(\mathrm { P } ( L < - 3.0\) or \(L > 3.0 )\).
A random sample of 20 rods cut by the machine was checked.
Find the probability that more than half of them were within 3.0 mm of the target length.
3. An estate agent sells properties at a mean rate of 7 per week.
Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model.
Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties.
Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties.
Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.
Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week.
Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.
Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly.
A manufacturer produces large quantities of coloured mugs. It is known from previous records that \(6 \%\) of the production will be green.
A random sample of 10 mugs was taken from the production line.
Define a suitable distribution to model the number of green mugs in this sample.
Find the probability that there were exactly 3 green mugs in the sample.
A random sample of 125 mugs was taken.
Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
a Poisson approximation,
a Normal approximation.
6. The continuous random variable \(X\) has probability density function
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 1 + x } { k } , & 1 \leqslant x \leqslant 4 \\
0 , & \text { otherwise }
\end{array} \right.$$
Show that \(k = \frac { 21 } { 2 }\).
Specify fully the cumulative distribution function of \(X\).
Calculate \(\mathrm { E } ( X )\).
Find the value of the median.
Write down the mode.
Explain why the distribution is negatively skewed.
It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.
Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to \(2.5 \%\) as possible.
State the actual significance level of the above test.
At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.
Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly.
Turn over
(a) Define a statistic.
A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { \mathrm { n } }\) is taken from a population with unknown mean \(\mu\).
For each of the following state whether or not it is a statistic.
\begin{enumerate}
\item It is estimated that $4 \%$ of people have green eyes. In a random sample of size $n$, the expected number of people with green eyes is 5 .\\
(a) Calculate the value of $n$.
\end{enumerate}
The expected number of people with green eyes in a second random sample is 3 .\\
(b) Find the standard deviation of the number of people with green eyes in this second sample. expected number of people with green eyes is 5 .\\
(a) Calculate the value of $n$ -
The expected number of people with green eyes in a second random sample is 3 .\\
(b) sample. C)
\begin{enumerate}
\item It is estimated that $4 \%$ of people have green eyes. In a random sample of size $n$, the\\
expected number of people with green eyes is 5 .\\
(a) Calculate the value of $n$.\\
The expected number of people with green eyes in a second random sample is 3 .\\
(b) Find the standard deviation of the number of people with green eyes in this second\\
sample.\\
\item The continuous random variable $X$ is uniformly distributed over the interval $[ 2,6 ]$.\\
(a) Write down the probability density function $\mathrm { f } ( x )$.
\end{enumerate}
Find\\
(b) $\mathrm { E } ( X )$,\\
(c) $\operatorname { Var } ( X )$,\\
(d) the cumulative distribution function of $X$, for all $x$,\\
(e) $\mathrm { P } ( 2.3 < X < 3.4 )$.\\
3. The random variable $X$ is the number of misprints per page in the first draft of a novel.\\
(a) State two conditions under which a Poisson distribution is a suitable model for $X$.
The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that\\
(b) a randomly chosen page has no misprints,\\
(c) the total number of misprints on 2 randomly chosen pages is more than 7 .
The first chapter contains 20 pages.\\
(d) Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.
4. Explain what you understand by\\
(a) a sampling unit,\\
(b) a sampling frame,\\
(c) a sampling distribution.\\
5. In a manufacturing process, $2 \%$ of the articles produced are defective. A batch of 200 articles is selected.\\
(a) Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.\\
(b) Estimate the probability there are less than 5 defective articles.\\
6. A continuous random variable $X$ has probability density function $\mathrm { f } ( x )$ where
$$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2 \\ 0 , & \text { otherwise } \end{cases}$$
where $k$ is a positive integer.\\
(a) Show that $k = \frac { 1 } { 4 }$.
Find\\
(b) $\mathrm { E } ( X )$,\\
(c) the mode of $X$,\\
(d) the median of $X$.\\
(e) Comment on the skewness of the distribution.\\
(f) Sketch f(x).\\
7. A drugs company claims that $75 \%$ of patients suffering from depression recover when treated with a new drug.
A random sample of 10 patients with depression is taken from a doctor's records.\\
(a) Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug.
Given that the claim is correct,\\
(b) find the probability that the treatment will be successful for exactly 6 patients.
The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.\\
(c) Stating your hypotheses clearly, test, at the $5 \%$ level of significance, the doctor's belief.\\
(d) From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the $1 \%$ level.
Turn over
\begin{enumerate}
\item Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.\\
(a) Explain why the secretary decided to take a random sample of club members rather than ask all the members.\\
(b) Suggest a suitable sampling frame.\\
(c) Identify the sampling units.\\
\includegraphics[max width=\textwidth, alt={}, center]{4d6588cd-22a7-4436-a7d9-9b335b98d2c0-014_90_72_2577_1805}\\
\includegraphics[max width=\textwidth, alt={}, center]{4d6588cd-22a7-4436-a7d9-9b335b98d2c0-014_104_1831_2648_114}
\item The continuous random variable $L$ represents the error, in mm , made when a machine cuts rods to a target length. The distribution of $L$ is continuous uniform over the interval [-4.0, 4.0].
\end{enumerate}
Find\\
(a) $\mathrm { P } ( L < - 2.6 )$,\\
(b) $\mathrm { P } ( L < - 3.0$ or $L > 3.0 )$.
A random sample of 20 rods cut by the machine was checked.\\
(c) Find the probability that more than half of them were within 3.0 mm of the target length.\\
3. An estate agent sells properties at a mean rate of 7 per week.\\
(a) Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model.\\
(b) Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties.\\
(c) Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties.
\begin{enumerate}
\setcounter{enumi}{3}
\item Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.\\
(a) Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week.
\end{enumerate}
Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.\\
(b) Test, at the $5 \%$ level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly.\\
\begin{enumerate}
\setcounter{enumi}{4}
\item A manufacturer produces large quantities of coloured mugs. It is known from previous records that $6 \%$ of the production will be green.
\end{enumerate}
A random sample of 10 mugs was taken from the production line.\\
(a) Define a suitable distribution to model the number of green mugs in this sample.\\
(b) Find the probability that there were exactly 3 green mugs in the sample.
A random sample of 125 mugs was taken.\\
(c) Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using\\
(i) a Poisson approximation,\\
(ii) a Normal approximation.\\
6. The continuous random variable $X$ has probability density function
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 1 + x } { k } , & 1 \leqslant x \leqslant 4 \\
0 , & \text { otherwise }
\end{array} \right.$$
(a) Show that $k = \frac { 21 } { 2 }$.\\
(b) Specify fully the cumulative distribution function of $X$.\\
(c) Calculate $\mathrm { E } ( X )$.\\
(d) Find the value of the median.\\
(e) Write down the mode.\\
(f) Explain why the distribution is negatively skewed.\\
\begin{enumerate}
\setcounter{enumi}{6}
\item It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.\\
(a) Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to $2.5 \%$ as possible.\\
(b) State the actual significance level of the above test.
\end{enumerate}
At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.\\
(c) Test, at the $10 \%$ level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly.
Turn over
\begin{enumerate}
\item (a) Define a statistic.
\end{enumerate}
A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { \mathrm { n } }$ is taken from a population with unknown mean $\mu$.\\
(b) For each of the following state whether or not it is a statistic.\\
(i) $\frac { X _ { 1 } + X _ { 4 } } { 2 }$,\\
(ii) $\frac { \sum X ^ { 2 } } { n } - \mu ^ { 2 }$.\\
\hfill \mbox{\textit{Edexcel S2 Q1}}