Questions S3 - A-Level Maths

use the Central Limit Theorem to find an approximate distribution for $\bar { Y }$ Given that the 60 observations of $Y$ have a sample mean of 13.4
find a $98 \%$ confidence interval for the maximum value that $Y$ can take.

Edexcel S3 2018 June Q7

7．（i）As part of a recruitment exercise candidates are required to complete three separate tasks．The times taken，$A , B$ and $C$ ，in minutes，for candidates to complete the three tasks are such that $$A \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 32,7 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 45,9 ^ { 2 } \right)$$ The time taken by an individual candidate to complete each task is assumed to be independent of the time taken to complete each of the other tasks． A candidate is selected at random．
（a）Find the probability that the candidate takes a total time of more than 90 minutes to complete all three tasks．
（b）Find $\mathrm { P } ( A > B )$
（ii）A simple random sample，$X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 4 }$ ，is taken from a normal population with mean $\mu$ and standard deviation $\sigma$ Given that $$\bar { X } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + X _ { 4 } } { 4 }$$ and that $$\mathrm { P } \left( X _ { 1 } > \bar { X } + k \sigma \right) = 0.1$$ where $k$ is a constant，
find the value of $k$ ，giving your answer correct to 3 significant figures．

END

Edexcel S3 2021 June Q1

A plant biologist claims that as the percentage moisture content of the soil in a field increases, so does the percentage plant coverage. He splits the field into equal areas labelled $A , B , C , D$ and $E$ and measures the percentage plant coverage and the percentage moisture content for each area. The results are shown in the table below.

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	$A$	$B$	$C$	$D$	$E$
Coverage \%	10	12	25	0	6
Moisture \%	30	20	40	10	25

Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test at the $5 \%$ level of significance, whether or not these data provide support for the plant biologist's claim.

Edexcel S3 2021 June Q2

A doctor believes that the diet of her patients and their health are not independent.

She takes a random sample of 200 patients and records whether they are in good health or poor health and whether they have a good diet or a poor diet. The results are summarised in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Good health	Poor health
Good diet	86	8
Poor diet	91	15

Stating your hypotheses clearly, test the doctor's belief using a $5 \%$ level of significance. Show your working for your test statistic and state your critical value clearly.

Edexcel S3 2021 June Q3

Components are manufactured such that their length in mm is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$. Below is a 95\% confidence interval for $\mu$ calculated from a random sample of components.
(11.52, 13.75)

Using the same random sample,

find a $90 \%$ confidence interval for $\mu$. Four 90\% confidence intervals are found from independent random samples.
Calculate the probability that only 3 of these 4 intervals will contain $\mu$.

Edexcel S3 2021 June Q4

A college runs academic and vocational courses. The college has 1680 academic students and 2520 vocational students.
1. Describe how a stratified sample of 70 students at the college could be taken.
All students at the college take a basic skills test. A random sample of 50 academic students has a mean score of 57 and a variance of 60. An independent random sample of 80 vocational students has a mean score of 62 with a variance of 70
Stating your hypotheses clearly, test at the $5 \%$ level of significance, whether or not the mean basic skills score for vocational students is greater than the mean basic skills score for academic students.
Explain the importance of the Central Limit Theorem to the test in part (b).
State an assumption that is required to carry out the test in part (b). All the academic students at the college take a basic skills course. Another random sample of 50 academic students and another independent random sample of 80 vocational students retake the basic skills test. The hypotheses used in part (b) are then tested again at the same level of significance. The value of the test statistic $z$ is now 1.54
Comment on the mean basic skills scores of academic and vocational students after taking this course.
Considering the outcomes of the tests in part (b) and part (e), comment on the effectiveness of the basic skills course.

Edexcel S3 2021 June Q5

A researcher is looking into the effectiveness of a new medicine for the relief of symptoms. He collects random samples of 8 people who are taking the medicine from each of 50 different medical practices. The number of people who say that the medicine is a success, in each sample, is recorded. The results are summarised in the table below.

Number of successes	0	1	2	3	4	5	6	7	8
Number of practices	4	6	3	12	10	7	4	2	2

The researcher decides to model this data using a binomial distribution.

State two necessary assumptions that the researcher made in order to use this model.
Show that the mean number of successes per sample is 3.54 He decides to use this mean to calculate expected frequencies. The results are shown in the table below.
Number of successes 0 1 2 3 4 5 6 7 8
Expected frequency 0.47 2.96 8.23 13.07 $f$ 8.23 3.27 0.74 $g$
Calculate the value of $f$ and the value of $g$. Give your answers to 2 decimal places.
Stating your hypotheses clearly, test at the $10 \%$ level of significance, whether or not the binomial distribution is a suitable model for the number of successes in samples of 8 people.

Edexcel S3 2021 June Q6

A baker produces bread buns and bread rolls. The weights of buns, $B$ grams, and the weights of rolls, $R$ grams, are such that $B \sim \mathrm {~N} \left( 55,1.3 ^ { 2 } \right)$ and $R \sim \mathrm {~N} \left( 51,1.2 ^ { 2 } \right)$

A bun and a roll are selected at random.

Find the probability that the bun weighs less than $110 \%$ of the weight of the roll. Two buns are chosen at random.
Find the probability that their weights differ by more than 1 gram. The baker sells bread in bags. Each bag contains either 10 buns or 11 rolls. The weight of an empty bag, $S$ grams, is such that $S \sim \mathrm {~N} \left( 3,0.2 ^ { 2 } \right)$
Find the probability that a bag of buns weighs less than a bag of rolls.

Edexcel S3 2022 June Q1

The table below shows the number of televised tournaments won and the total number of tournaments won by the top 10 ranked darts players in 2020

Player's rank	Televised tournaments won	Total tournaments won
1	55	135
2	7	33
3	5	17
4	2	14
5	4	9
6	2	5
7	9	36
8	0	15
9	3	3
10	0	13

Michael did not want to calculate Spearman’s rank correlation coefficient between player's rank and the rank in televised tournaments won because there would be tied ranks.

Explain how Michael could have dealt with these tied ranks. Given that the largest number of total tournaments won is ranked number 1
calculate the value of Spearman's rank correlation coefficient between player's rank and the rank in total tournaments won.
Stating your hypotheses and critical value clearly, test at the $5 \%$ level of significance, whether or not there is evidence of a positive correlation between player's rank and the rank in total tournaments won for these darts players. Michael does not believe that there is a positive correlation between player's rank and the rank in total number of tournaments won.
Find the largest level of significance, that is given in the tables provided, which could be used to support Michael's claim.
You must state your critical value.

Edexcel S3 2022 June Q2

An experiment is conducted to compare the heat retention of two brands of flasks, brand $A$ and brand $B$. Both brands of flask have a capacity of 750 ml .

In the experiment 750 ml of boiling water is poured into the flask, which is then sealed. Four hours later the temperature, in ${ } ^ { \circ } \mathrm { C }$, of the water in the flask is recorded. A random sample of 100 flasks from brand $A$ gives the following summary statistics, where $x$ is the temperature of the water in the flask after four hours. $$\sum x = 7690 \quad \sum ( x - \bar { x } ) ^ { 2 } = 669.24$$

Find unbiased estimates for the mean and variance of the temperature of the water, after four hours, for brand $A$. A random sample of 80 flasks from brand $B$ gives the following results, where $y$ is the temperature of the water in the flask after four hours. $$\bar { y } = 75.9 \quad s _ { y } = 2.2$$
Test, at the $1 \%$ significance level, whether there is a difference in the mean water temperature after four hours between brand $A$ and brand $B$. You should state your hypotheses, test statistic and critical value clearly.
Explain why it is reasonable to assume that $\sigma ^ { 2 } = s ^ { 2 }$ in this situation.

Edexcel S3 2022 June Q3

The random variable $X$ is normally distributed with unknown mean $\mu$ and known variance $\sigma ^ { 2 }$

A random sample of 25 observations of $X$ produced a $95 \%$ confidence interval for $\mu$ of (26.624, 28.976)

Find the mean of the sample.
Show that the standard deviation is 3 The $a$ \% confidence interval using the 25 observations has a width of 2.1
Calculate the value of $a$
Find the smallest sample size, of observations from $X$, that would be required to obtain a 95\% confidence interval of width at most 1.5

Edexcel S3 2022 June Q4

Navtej travels to work by train. A train leaves the station every 7 minutes and Navtej's arrival at the station is independent of when the train is due to leave.
1. Write down a suitable model for the distribution of the time, $T$ minutes, that he has to wait for a train to leave.
2. Find the mean and standard deviation of $T$
During a 10-week period, Navtej travels to work by train on 46 occasions.
Estimate the probability that the mean length of time that he has to wait for a train to leave is between 3.4 and 3.6 minutes.
State a necessary assumption for the calculation in part (c).

Edexcel S3 2022 June Q5

A random sample of two observations $X _ { 1 }$ and $X _ { 2 }$ is taken from a population with unknown mean $\mu$ and unknown variance $\sigma ^ { 2 }$
1. Explain why $\frac { X _ { 1 } - \mu } { \sigma }$ is not a statistic.
2. Explain what you understand by an unbiased estimator for $\mu$
Two estimators for $\mu$ are $U _ { 1 }$ and $U _ { 2 }$ where $$U _ { 1 } = 3 X _ { 1 } - 2 X _ { 2 } \quad \text { and } \quad U _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } } { 4 }$$
Show that both $U _ { 1 }$ and $U _ { 2 }$ are unbiased estimators for $\mu$ The most efficient estimator among a group of unbiased estimators is the one with the smallest variance.
By finding the variance of $U _ { 1 }$ and the variance of $U _ { 2 }$ state, giving a reason, the most efficient estimator for $\mu$ from these two estimators.

Edexcel S3 2022 June Q6

6 A particular lift has a maximum load capacity of 700 kg .
The weights of men are normally distributed with mean 80 kg and standard deviation 10 kg . The weights of women are normally distributed with mean 69 kg and standard deviation 5 kg . You may assume that weights of people are independent.

Find the probability that when 6 men and 3 women are in the lift, the load exceeds 700 kg . A sign in the lift states: "Maximum number of people in the lift is $c$ "
Find the value of $c$ such that the probability of the load exceeding 700 kg is less than $2.5 \%$ no matter the gender of the occupants.

Edexcel S3 2022 June Q7

7 The following table shows observed frequencies, where $x$ is an integer, from an experiment to test whether or not a six-sided die is biased.

Number on die	1	2	3	4	5	6
Observed frequency	$x + 6$	$x - 8$	$x + 8$	$x - 5$	$x + 4$	$x - 5$

A goodness of fit test is conducted to determine if there is evidence that the die is biased.

Write down suitable null and alternative hypotheses for this test. It is found that the null hypothesis is not rejected at the $5 \%$ significance level.
Hence
1. find the minimum value of $x$
2. determine the minimum number of times the die was rolled.

Edexcel S3 2023 June Q1

(a) State two conditions under which it might be more appropriate to use Spearman's rank correlation coefficient rather than the product moment correlation coefficient.

A random sample of 10 melons was taken from a market stall. The length, in centimetres, and maximum diameter, in centimetres, of each melon were recorded. The Spearman's rank correlation coefficient between the results was - 0.673
(b) Test, at the $5 \%$ level of significance, whether or not there is evidence of a correlation. State clearly your hypotheses and the critical value used. The product moment correlation coefficient between the results was - 0.525
(c) Test, at the $5 \%$ level of significance, whether or not there is evidence of a negative correlation.
State clearly your hypotheses and the critical value used.

Edexcel S3 2023 June Q2

A business accepts cash, bank cards or mobile apps as payment methods.

The manager wishes to test whether or not there is an association between the payment amount and the payment method used. The manager takes a random sample of 240 payments and records the payment amount and the payment method used. The manager's results are shown in the table.

\multirow{2}{*}{}		Payment amount
		Under £50	£50 to £150	Over £150
\multirow{3}{*}{Payment method}	Cash	23	19	18
	Bank card	21	32	31
	Mobile app	16	39	41

Using these results,

calculate the expected frequencies for the payment amount under $\pounds 50$ that
1. use cash
2. use a bank card
3. use a mobile app Given that for the other 6 classes $\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.4048$ to 4 decimal places,
test, at the $5 \%$ level of significance, whether or not there is evidence for an association between the payment amount and the payment method used. You should state the hypotheses, the test statistic, the degrees of freedom and the critical value used for this test.

Edexcel S3 2023 June Q3

A random sample of 2 observations, $X _ { 1 }$ and $X _ { 2 }$, is taken from a population with unknown mean $\mu$ and unknown variance $\sigma ^ { 2 }$
1. Explain why $\frac { X _ { 1 } - X _ { 2 } } { \sigma }$ is not a statistic.
$$S = \frac { 3 } { 5 } X _ { 1 } + \frac { 5 } { 7 } X _ { 2 }$$
Show that $S$ is a biased estimator of $\mu$
Hence find the bias, in terms of $\mu$, when $S$ is used as an estimator of $\mu$ Given that $Y = a X _ { 1 } + b X _ { 2 }$ is an unbiased estimator of $\mu$, where $a$ and $b$ are constants,
find an equation, in terms of $a$ and $b$, that must be satisfied.
Using your answer to part (d), show that $\operatorname { Var } ( Y ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }$

Edexcel S3 2023 June Q4

It is suggested that the delay, in hours, of certain flights from a particular country may be modelled by the continuous random variable, $T$, with probability density function

$$f ( t ) = \left\{ \begin{array} { c l } \frac { 2 } { 25 } t & 0 \leqslant t < 5
0 & \text { otherwise } \end{array} \right.$$

Show that for $0 \leqslant a \leqslant 4$ $$P ( a \leqslant T < a + 1 ) = \frac { 1 } { 25 } ( 2 a + 1 )$$ A random sample of 150 of these flights is taken. The delays are summarised in the table below.
Delay ( $\boldsymbol { t }$ hours) Frequency
$0 \leqslant t < 1$ 10
$1 \leqslant t < 2$ 13
$2 \leqslant t < 3$ 24
$3 \leqslant t < 4$ 35
$4 \leqslant t < 5$ 68
Test, at the $5 \%$ significance level, whether the given probability density function is a suitable model for these delays.
You should state your hypotheses, expected frequencies, test statistic and the critical value used.

Edexcel S3 2023 June Q5

The continuous random variable $X$ is normally distributed with

$$X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)$$ A random sample of 10 observations of $X$ is taken and $\bar { X }$ denotes the sample mean.

Show that a $90 \%$ confidence interval for $\mu$, in terms of $\bar { x }$, is given by $$( \bar { x } - 2.60 , \bar { x } + 2.60 )$$ The continuous random variable $Y$ is normally distributed with $$Y \sim \mathrm {~N} \left( \mu , 3 ^ { 2 } \right)$$ A random sample of 20 observations of $Y$ are taken and $\bar { Y }$ denotes the sample mean.
Find a 95\% confidence interval for $\mu$, in terms of $\bar { y }$
Given that $X$ and $Y$ are independent,
1. find the distribution of $\bar { X } - \bar { Y }$
2. calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.

Edexcel S3 2023 June Q6

Roxane, a scientist, carries out an investigation into the fat content of different brands of crisps.

Roxane took random samples of different brands of crisps and recorded, in grams, the fat content ( $x$ ) of a 30 gram serving. The table below shows some results for just two of these brands.

Brand	$\sum x$	$\sum \boldsymbol { x } ^ { \mathbf { 2 } }$	$\bar { x }$	$s$	Sample size
A	350	1753.9744	5.0	0.24	70
B	331.5	1694.65	$\alpha$	β	65

Calculate the value of $\alpha$ and the value of $\beta$ Roxane claims that these results show that the crisps from brand A have a lower fat content than the crisps from brand B , as the mean fat content for brand A is, statistically, significantly less than the mean fat content for brand B .
Stating your hypotheses clearly, carry out a suitable test, at the $5 \%$ level of significance, to assess Roxane's claim.
You should state your test statistic and critical value.
For the test in part (b), state whether or not it is necessary to assume that the fat content of crisps is normally distributed. Give a reason for your answer.
State an assumption you have made in carrying out the test in part (b).

Edexcel S3 2023 June Q7

The random variable $X$ is defined as

$$X = 4 A - 3 B$$ where $A$ and $B$ are independent and $$A \sim \mathrm {~N} \left( 15,5 ^ { 2 } \right) \quad B \sim \mathrm {~N} \left( 10,4 ^ { 2 } \right)$$

Find $\mathrm { P } ( X < 40 )$ The random variable $C$ is such that $C \sim \mathrm {~N} \left( 20 , \sigma ^ { 2 } \right)$
The random variables $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ are independent and each has the same distribution as $C$ The random variable $D$ is defined as $$D = \sum _ { i = 1 } ^ { 3 } C _ { i }$$ Given that $\mathrm { P } ( A + B + D < 76 ) = 0.2420$ and that $A , B$ and $D$ are independent,
showing your working clearly, find the standard deviation of $C$

Edexcel S3 2024 June Q1

The names of the 400 employees of a company are listed alphabetically in a book.

The chairperson of the company wishes to select a sample of 8 employees.
The chairperson numbers the employees from 001 to 400

Describe how the list of numbers can be used to select a systematic sample of 8 employees.
State one disadvantage of systematic sampling in this case.
Write down the probability that the sample includes both the first name (employee 001) and the last name (employee 400) in the list.

Edexcel S3 2024 June Q2

Aarush is asked to estimate the price of 7 kettles and rank them in order of decreasing price.

Aarush's order of decreasing price is $D A F C B G E$
The actual prices of the 7 kettles are shown in the table below.

Kettle	$A$	$B$	$C$	$D$	$E$	$F$	$G$
Price (£)	99.99	14.99	34.97	49.99	19.97	29.99	8.99

Calculate Spearman's rank correlation coefficient between Aarush's order and the actual order. Use a rank of 1 for the highest priced kettle.
Show your working clearly.
Using a $5 \%$ level of significance, test whether or not there is evidence to suggest that Aarush is able to rank kettles in order of decreasing price. You should state your hypotheses and critical value.
Explain why Aarush did not use the product moment correlation coefficient in this situation. Aarush discovered that kettle A's price was recorded incorrectly and should have been $\pounds 49.99$ rather than $\pounds 99.99$
Explain what effect this has on the rankings for the price.

Edexcel S3 2024 June Q3

The volume of water in a bottle has a normal distribution with unknown mean, $\mu$ millilitres, and known standard deviation, $\sigma$ millilitres.

A random sample of 150 of the bottles of water gave a 95\% confidence interval for $\mu$ of
(327.84, 329.76)

Using the confidence interval given, test whether or not $\mu = 328$ State your hypotheses clearly and write down the significance level you have used. A second random sample, of 200 of these bottles of water, had a mean volume of 328 millilitres.
Calculate a 98\% confidence interval for $\mu$ based on this second sample. You must show all steps in your working.
(Solutions relying entirely on calculator technology are not acceptable.) Using five different random samples of 200 of these bottles of water, five $98 \%$ confidence intervals for $\mu$ are to be found.
Calculate the probability that more than 3 of these intervals will contain $\mu$

Number on die	1	2	3	4	5	6
Observed frequency	\(x + 6\)	\(x - 8\)	\(x + 8\)	\(x - 5\)	\(x + 4\)	\(x - 5\)

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
Coverage \%	10	12	25	0	6
Moisture \%	30	20	40	10	25

Number of successes	0	1	2	3	4	5	6	7	8
Expected frequency	0.47	2.96	8.23	13.07	\(f\)	8.23	3.27	0.74	\(g\)

Delay ( \(\boldsymbol { t }\) hours)	Frequency
\(0 \leqslant t < 1\)	10
\(1 \leqslant t < 2\)	13
\(2 \leqslant t < 3\)	24
\(3 \leqslant t < 4\)	35
\(4 \leqslant t < 5\)	68

Brand	\(\sum x\)	\(\sum \boldsymbol { x } ^ { \mathbf { 2 } }\)	\(\bar { x }\)	\(s\)	Sample size
A	350	1753.9744	5.0	0.24	70
B	331.5	1694.65	\(\alpha\)	β	65

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