Question 7 - A-Level Maths

Edexcel S3 — Question 7

Exam Board	Edexcel
Module	S3 (Statistics 3)
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear combinations of normal random variables
Type	Comparison involving sums or multiples
Difficulty	Standard +0.8 This question requires understanding of linear combinations of normal distributions, including forming new distributions (L - 4S and 4S), working with sums of independent normals, and applying standardization. Part (c) involves absolute values requiring probability manipulation. While the techniques are standard for S3, the multi-step reasoning and combination of concepts (independence, variance rules, absolute value probabilities) elevates this above routine exercises.
Spec	5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions

7. A set of scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution $\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)$. The random variables $L$ and $S$ are independent. A long pole and a short pole are selected at random.

Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable $T$ represents the length of the row.
Find the distribution of $T$.
Find $\mathrm { P } ( | L - T | < 0.1 )$.
\end{table}
1. Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, $x \mathrm {~mm}$ of each wading bird was recorded. The results are summarised as follows
$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$
Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
find a $99 \%$ confidence interval for the mean wing length of the birds from this group.
2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results \end{table}
1. A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
2. Explain in detail how the researcher should obtain such a sample.
3. Give one advantage and one disadvantage of
  1. quota sampling,
  2. systematic sampling.
4. The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm . The heights of the orchids are normally distributed.
Given that the population standard deviation is 0.5 cm ,
estimate limits between which $95 \%$ of the heights of the orchids lie,
find a 98\% confidence interval for the mean height of the orchids. A grower claims that the mean height of this type of orchid is 19.5 cm .
Comment on the grower's claim. Give a reason for your answer.
3. A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.
Individual $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
BMI 17.4 21.4 18.9 24.4 19.4 20.1 22.6 18.4 25.8 28.1
Finishing position 3 5 1 9 6 4 10 2 7 8
Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly and using a one tailed test with a $5 \%$ level of significance, interpret your rank correlation coefficient.
Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.
4. A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.
5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below. \begin{table}[h]
Number of goals Frequency
0 40
1 33
2 14
3 8
4 5
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
Calculate the mean number of goals scored per game. The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]
Number of goals Expected Frequency
0 34.994
1 $r$
2 $s$
3 6.752
$\geqslant 4$ 2.221
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Find the value of $r$ and the value of $s$ giving your answers to 3 decimal places.
Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim.
1. The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm . The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm .
2. Test, using a $5 \%$ level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.
3. State two assumptions you made in carrying out the test in part (a).
4. A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below.
  119.9
  120.3
  120.1
  120.4
  120.2
5. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.
The lengths of climbing rope are known to have a standard deviation of 0.2 m . The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of $n$, lies within 0.05 m of its true value.
Find the minimum sample size required.

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7. A set of scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution $\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)$. The random variables $L$ and $S$ are independent.

A long pole and a short pole are selected at random.\\
(a) Find the probability that the length of the long pole is more than 4 times the length of the short pole.

Four short poles are selected at random and placed end to end in a row. The random variable $T$ represents the length of the row.\\
(b) Find the distribution of $T$.\\
(c) Find $\mathrm { P } ( | L - T | < 0.1 )$.\\

\end{table}

\begin{enumerate}
  \item Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, $x \mathrm {~mm}$ of each wading bird was recorded. The results are summarised as follows
\end{enumerate}

$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$

(a) Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds.

Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,\\
(b) find a $99 \%$ confidence interval for the mean wing length of the birds from this group.\\

2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results

\end{table}

\begin{enumerate}
  \item A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.\\
(a) Explain in detail how the researcher should obtain such a sample.\\
(b) Give one advantage and one disadvantage of\\
(i) quota sampling,\\
(ii) systematic sampling.\\

  \item The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm . The heights of the orchids are normally distributed.
\end{enumerate}

Given that the population standard deviation is 0.5 cm ,\\
(a) estimate limits between which $95 \%$ of the heights of the orchids lie,\\
(b) find a 98\% confidence interval for the mean height of the orchids.

A grower claims that the mean height of this type of orchid is 19.5 cm .\\
(c) Comment on the grower's claim. Give a reason for your answer.\\

3. A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Individual & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ \\
\hline
BMI & 17.4 & 21.4 & 18.9 & 24.4 & 19.4 & 20.1 & 22.6 & 18.4 & 25.8 & 28.1 \\
\hline
Finishing position & 3 & 5 & 1 & 9 & 6 & 4 & 10 & 2 & 7 & 8 \\
\hline
\end{tabular}
\end{center}

(a) Calculate Spearman's rank correlation coefficient for these data.\\
(b) Stating your hypotheses clearly and using a one tailed test with a $5 \%$ level of significance, interpret your rank correlation coefficient.\\
(c) Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.\\
4. A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.\\
5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below.

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline
Number of goals & Frequency \\
\hline
0 & 40 \\
\hline
1 & 33 \\
\hline
2 & 14 \\
\hline
3 & 8 \\
\hline
4 & 5 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}

(a) Calculate the mean number of goals scored per game.

The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2.

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline
Number of goals & Expected Frequency \\
\hline
0 & 34.994 \\
\hline
1 & $r$ \\
\hline
2 & $s$ \\
\hline
3 & 6.752 \\
\hline
$\geqslant 4$ & 2.221 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{center}
\end{table}

(b) Find the value of $r$ and the value of $s$ giving your answers to 3 decimal places.\\
(c) Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim.\\

\begin{enumerate}
  \item The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm . The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm .\\
(a) Test, using a $5 \%$ level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.\\
(b) State two assumptions you made in carrying out the test in part (a).\\

  \item A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below.\\
119.9\\
120.3\\
120.1\\
120.4\\
120.2\\
(a) Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.
\end{enumerate}

The lengths of climbing rope are known to have a standard deviation of 0.2 m . The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of $n$, lies within 0.05 m of its true value.\\
(b) Find the minimum sample size required.\\

\hfill \mbox{\textit{Edexcel S3  Q7}}

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Q7 Q8

Individual	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
BMI	17.4	21.4	18.9	24.4	19.4	20.1	22.6	18.4	25.8	28.1
Finishing position	3	5	1	9	6	4	10	2	7	8

Number of goals	Expected Frequency
0	34.994
1	\(r\)
2	\(s\)
3	6.752
\(\geqslant 4\)	2.221