Single tail probability P(X < a) or P(X > a)

CAIE S1 2013 November Q1

3 marks Easy -1.2

1 It is given that $X \sim \mathrm {~N} \left( 1.5,3.2 ^ { 2 } \right)$. Find the probability that a randomly chosen value of $X$ is less than - 2.4 .

Edexcel S1 2014 January Q6

9 marks Moderate -0.3

6. A manufacturer has a machine that fills bags with flour such that the weight of flour in a bag is normally distributed. A label states that each bag should contain 1 kg of flour.

The machine is set so that the weight of flour in a bag has mean 1.04 kg and standard deviation 0.17 kg . Find the proportion of bags that weigh less than the stated weight of 1 kg . The manufacturer wants to reduce the number of bags which contain less than the stated weight of 1 kg . At first she decides to adjust the mean but not the standard deviation so that only $5 \%$ of the bags filled are below the stated weight of 1 kg .
Find the adjusted mean. The manufacturer finds that a lot of the bags are overflowing with flour when the mean is adjusted, so decides to adjust the standard deviation instead to make the machine more accurate. The machine is set back to a mean of 1.04 kg . The manufacturer wants $1 \%$ of bags to be under 1 kg .
Find the adjusted standard deviation. Give your answer to 3 significant figures.

Edexcel Paper 3 2024 June Q5

10 marks Standard +0.3

The records for a school athletics club show that the height, $H$ metres, achieved by students in the high jump is normally distributed with mean 1.4 metres and standard deviation 0.15 metres.
1. Find the proportion of these students achieving a height of more than 1.6 metres.
The records also show that the time, $T$ seconds, to run 1500 metres is normally distributed with mean 330 seconds and standard deviation 26 seconds. The school's Head would like to use these distributions to estimate the proportion of students from the school athletics club who can jump higher than 1.6 metres and can run 1500 metres in less than 5 minutes.
State a necessary assumption about $H$ and $T$ for the Head to calculate an estimate of this proportion.
Find the Head's estimate of this proportion. Students in the school athletics club also throw the discus.
The random variable $D \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$ represents the distance, in metres, that a student can throw the discus. Given that $\mathrm { P } ( D < 16.3 ) = 0.30$ and $\mathrm { P } ( D > 29.0 ) = 0.10$
calculate the value of $\mu$ and the value of $\sigma$

Edexcel S2 2005 January Q1

4 marks Easy -1.2

The random variables $R , S$ and $T$ are distributed as follows

$$R \sim \mathrm {~B} ( 15,0.3 ) , \quad S \sim \mathrm { Po } ( 7.5 ) , \quad T \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) .$$ Find

$\mathrm { P } ( R = 5 )$,
$\mathrm { P } ( S = 5 )$,
$\mathrm { P } ( T = 5 )$.

Edexcel S1 Q4

10 marks Standard +0.3

4. A company produces jars of English Honey. The weight of the glass jars used is normally distributed with a mean of 122.3 g and a standard deviation of 2.6 g . Calculate the probability that a randomly chosen jar will weigh

less than 127 g ,
less than 121.5 g . The weight of honey put into each jar by a machine is normally distributed with a standard deviation of 1.6 g . The machine operator can adjust the mean weight of the honey put into each jar without changing the standard deviation.
Find, correct to 4 significant figures, the minimum that the mean weight can be set to such that at most 1 in 20 of the jars will contain less than 454 g .
(4 marks)

OCR S2 2007 June Q4

6 marks Moderate -0.3

State two conditions needed for $X$ to be well modelled by a normal distribution.
It is given that $X \sim \mathrm {~N} \left( 50.0,8 ^ { 2 } \right)$. The mean of 20 random observations of $X$ is denoted by $\bar { X }$. Find $\mathrm { P } ( \bar { X } > 47.0 )$. 5 The number of system failures per month in a large network is a random variable with the distribution $\operatorname { Po } ( \lambda )$. A significance test of the null hypothesis $\mathrm { H } _ { 0 } : \lambda = 2.5$ is carried out by counting $R$, the number of system failures in a period of 6 months. The result of the test is that $\mathrm { H } _ { 0 }$ is rejected if $R > 23$ but is not rejected if $R \leqslant 23$.
State the alternative hypothesis.
Find the significance level of the test.
Given that $\mathrm { P } ( R > 23 ) < 0.1$, use tables to find the largest possible actual value of $\lambda$. You should show the values of any relevant probabilities. 6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter $e$ appears $19 \%$ of the time. A certain encoded message of 20 letters contains one letter $e$.
Using an exact binomial distribution, test at the $10 \%$ significance level whether there is evidence that the proportion of the letter $e$ in the language from which this message is a sample is less than in German, i.e., less than $19 \%$.
Give a reason why a binomial distribution might not be an appropriate model in this context. 7 Two continuous random variables $S$ and $T$ have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
Sketch on the same axes the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$. [You should not use graph paper or attempt to plot points exactly.]
Explain in everyday terms the difference between the two random variables.
Find the value of $t$ such that $\mathrm { P } ( T > t ) = 0.2$. 8 A random variable $Y$ is normally distributed with mean $\mu$ and variance 12.25. Two statisticians carry out significance tests of the hypotheses $\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0$.
Statistician $A$ uses the mean $\bar { Y }$ of a sample of size 23, and the critical region for his test is $\bar { Y } > 64.20$. Find the significance level for $A$ 's test.
Statistician $B$ uses the mean of a sample of size 50 and a significance level of $5 \%$.
1. Find the critical region for $B$ 's test.
2. Given that $\mu = 65.0$, find the probability that $B$ 's test results in a Type II error.
3. Given that, when $\mu = 65.0$, the probability that $A$ 's test results in a Type II error is 0.1365 , state with a reason which test is better. 9 (a) The random variable $G$ has the distribution $\mathrm { B } ( n , 0.75 )$. Find the set of values of $n$ for which the distribution of $G$ can be well approximated by a normal distribution.
  (b) The random variable $H$ has the distribution $\mathrm { B } ( n , p )$. It is given that, using a normal approximation, $\mathrm { P } ( H \geqslant 71 ) = 0.0401$ and $\mathrm { P } ( H \leqslant 46 ) = 0.0122$.
  1. Find the mean and standard deviation of the approximating normal distribution.
  2. Hence find the values of $n$ and $p$.

CAIE S1 2021 November Q8

Easy -1.8

8	MATHEMATICS	9709/52
0	Paper 5 Probability \	Statistics 1	October/November 2021
$\infty$		1 hour 15 minutes
	You must answer on the question paper.
	You will need: List of formulae (MF19)

\section*{INSTRUCTIONS}

Answer all questions.
Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
Write your name, centre number and candidate number in the boxes at the top of the page.
Write your answer to each question in the space provided.
Do not use an erasable pen or correction fluid.
Do not write on any bar codes.
If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown.
You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

\section*{INFORMATION}

The total mark for this paper is 50.
The number of marks for each question or part question is shown in brackets [ ].

1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.

	Piano	Guitar	Drums
Male	25	44	11
Female	42	38	20

A student at the college is chosen at random.

Find the probability that the student plays the guitar.
Find the probability that the student is male given that the student plays the drums.
Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
2 A group of 6 people is to be chosen from 4 men and 11 women.
1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
  Two of the 11 women are sisters Jane and Kate.
2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
  3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
3. Show that the probability that exactly one of the marbles is yellow is $\frac { 5 } { 14 }$.
  The random variable $X$ is the number of yellow marbles selected.
4. Draw up the probability distribution table for $X$.
5. Find $\mathrm { E } ( X )$.
  4
6. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
7. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
  5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.
8. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
9. Find the probability that the first wet day in October is 8 October.
10. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.
  6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.
11. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
12. $20 \%$ of employees take longer than $t$ minutes to complete the task. Find the value of $t$.
13. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.
  7 The distances, $x \mathrm {~m}$, travelled to school by 140 children were recorded. The results are summarised in the table below.
  Distance, $x \mathrm {~m}$ $x \leqslant 200$ $x \leqslant 300$ $x \leqslant 500$ $x \leqslant 900$ $x \leqslant 1200$ $x \leqslant 1600$
  Cumulative frequency 16 46 88 122 134 140
14. On the grid, draw a cumulative frequency graph to represent these results. \includegraphics[max width=\textwidth, alt={}, center]{93ff111b-0267-4b4b-a41c-64c3307115af-10_1593_1593_701_306}
15. Use your graph to estimate the interquartile range of the distances.
16. Calculate estimates of the mean and standard deviation of the distances.
  If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

AQA S1 2005 January Q2

9 marks Moderate -0.3

2 The volume, in millilitres, of lemonade in mini-cans may be assumed to be normally distributed with a standard deviation of 3.5. The volumes, in millilitres, of lemonade in a random sample of 12 mini-cans were as follows.

155	148	156	149	147	156
157	156	150	154	148	154

Construct a $98 \%$ confidence interval for the mean volume of lemonade in a mini-can, giving the limits to one decimal place.
On each mini-can is printed " 150 ml ". Comment on this, using the given sample and your confidence interval in part (a).
State why, in part (a), use of the Central Limit Theorem was not necessary.

AQA S1 2006 June Q4

7 marks Moderate -0.3

4 The weights of packets of sultanas may be assumed to be normally distributed with a standard deviation of 6 grams. The weights of a random sample of 10 packets were as follows: $\begin{array} { l l l l l l l l l l } 498 & 496 & 499 & 511 & 503 & 505 & 510 & 509 & 513 & 508 \end{array}$

1. Construct a $99 \%$ confidence interval for the mean weight of packets of sultanas, giving the limits to one decimal place.
2. State why, in calculating your confidence interval, use of the Central Limit Theorem was not necessary.
3. On each packet it states 'Contents 500 grams'. Comment on this statement using both the given sample and your confidence interval.
Given that the mean weight of all packets of sultanas is 500 grams, state the probability that a 99\% confidence interval for the mean, calculated from a random sample of packets, will not contain 500 grams.

AQA Further Paper 3 Statistics 2019 June Q3

4 marks Standard +0.8

3 Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes. 3

Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place.
3
Alan claims that his mean journey time to work is 30 minutes.
State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim.
3
Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6 Explain how the working in part (a) would change.

Pre-U Pre-U 9794/3 2012 June Q2

5 marks Moderate -0.8

2 A bag contains four black balls and one white ball. A man chooses a ball at random. If it is a black ball, he replaces it and chooses another at random. If he chooses the white ball, he stops.

Name the probability distribution which models this situation.
Calculate the probability that he will make exactly three attempts before he stops.
Calculate the probability that he will make fewer than three attempts before he stops.

Distance, \(x \mathrm {~m}\)	\(x \leqslant 200\)	\(x \leqslant 300\)	\(x \leqslant 500\)	\(x \leqslant 900\)	\(x \leqslant 1200\)	\(x \leqslant 1600\)
Cumulative frequency	16	46	88	122	134	140