Browse Statistical distributions

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Discrete random variables and probability distributions

39 questions

LessonNot started
A discrete random variable XX has P(X=x)=kxP(X = x) = \dfrac{k}{x} for x=1,2,3,4x = 1, 2, 3, 4. Find kk, to 3 s.f.
●●●●●Level 32 marksStart
A discrete random variable XX has P(X=x)=cxP(X = x) = cx for x=1,2,3,4x = 1, 2, 3, 4. Find the value of cc.
●●●●●Level 32 marksStart
The cumulative distribution function of XX satisfies F(2)=0.5F(2) = 0.5 and F(3)=0.8F(3) = 0.8. Find P(X=3)P(X = 3).
●●●●●Level 32 marksStart
The discrete random variable XX has distribution

x0123P(X=x)a0.30.2b\begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \\\\ \hline P(X=x) & a & 0.3 & 0.2 & b \end{array}

Given that P(X≤1)=0.5P(X \le 1) = 0.5, find aa and bb.
●●●●●Level 33 marksStart
The discrete random variable XX has distribution

x12345P(X=x)0.20.30.250.150.1\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\\\ \hline P(X=x) & 0.2 & 0.3 & 0.25 & 0.15 & 0.1 \end{array}

Find P(X≥3)P(X \ge 3).
●●●●●Level 32 marksStart
SECTION B: STATISTICS

The discrete random variable XX has probability distribution

P(X=x)=kxP(X = x) = kx, x=1,2,3,4,5\quad x = 1, 2, 3, 4, 5,

where kk is a constant.
(a) Show that k=115k = \dfrac{1}{15}. (2 marks)
(b) Find P(X≥4)P(X \ge 4). (2 marks)
(c) Find P(X≥4∣X≥2)P(X \ge 4 \mid X \ge 2). (2 marks)
●●●●●Level 36 marksStart
The discrete random variable XX has the probability distribution shown.

x1234P(X=x)k2k3k4k\begin{array}{|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(X = x) & k & 2k & 3k & 4k \\ \hline \end{array}
(a) Find the value of kk. (2 marks)
(b) Find E(X)\mathrm{E}(X). (2 marks)
(c) Find Var(X)\mathrm{Var}(X). (3 marks)
(d) Find P(X>E(X))\mathrm{P}\left(X > \mathrm{E}(X)\right). (1 mark)
●●●●●Level 38 marksStart
A builders' merchant has a very large stock of paving slabs. Each slab is either 4040 cm long or 6060 cm long, and 4040 cm slabs and 6060 cm slabs are in the ratio 7:37 : 3 respectively.

A random sample of 33 slabs is taken from the stock and the slabs are laid end to end in a row. The random variable TT represents the total length, in cm, of the row.
(a) Find the sampling distribution of TT

(6 marks)
(b) Find P(T>150)P(T > 150)

(1 mark)
●●●●●Level 37 marksStart
A fair six-sided die is rolled once and XX is the score. State the name of the distribution of XX, and find P(X>4)P(X > 4).
●●●●●Level 32 marksStart
A discrete random variable XX has P(X=x)=kx2P(X = x) = kx^2 for x=1,2,3x = 1, 2, 3.
(a) Find kk.
(b) Find P(X=3)P(X = 3).
●●●●●Level 33 marksStart
The discrete random variable XX has the probability distribution shown in the table below.
xx113344
P(X=x)P(X = x)0.20.20.50.50.30.3
A random sample of 22 independent observations of XX is taken, and Xˉ\bar{X} is the mean of the 22 observations.
(a) Find the sampling distribution of Xˉ\bar{X}

(5 marks)
(b) Find P(Xˉ>μ)P(\bar{X} > \mu), where μ=E(X)\mu = E(X)

(2 marks)
●●●●●Level 37 marksStart
A garden centre buys trays of bedding plants from two suppliers, AA and BB. Each tray holds either 1515 plants or 2020 plants.
  • For supplier AA, trays of 1515 plants and trays of 2020 plants are in the ratio 2:32 : 3 respectively
  • For supplier BB, trays of 1515 plants and trays of 2020 plants are in the ratio 3:13 : 1 respectively
One tray is selected at random from a very large delivery from supplier AA, and two trays are selected at random from a very large delivery from supplier BB. The random variable TT represents the total number of plants on the 33 trays.
(a) Show that P(T=50)=3980P(T = 50) = \frac{39}{80}

(3 marks)
(b) Find the sampling distribution of TT

(3 marks)
●●●●●Level 36 marksStart
A game uses a bag containing a very large number of discs. Each disc shows the number 11, 22 or 44, and discs showing 11, 22 and 44 are in the ratio 2:1:12 : 1 : 1 respectively.

A player takes 22 discs at random from the bag, one after the other. The player's score, SS, is 33 times the number on the first disc plus the number on the second disc.
(a) Show that P(S=7)=14P(S = 7) = \frac14

(2 marks)
(b) Find the sampling distribution of SS

(4 marks)
●●●●●Level 36 marksStart
A garden centre sells a very large number of packets of seeds. Each packet contains either 2020 seeds or 5050 seeds. The proportion of packets containing 2020 seeds is pp, where p>0.5p > 0.5

A random sample of 33 packets is taken. The probability that all 33 packets contain the same number of seeds is 0.280.28
(a) Show that p2−p+0.24=0p^2 - p + 0.24 = 0

(2 marks)
(b) Hence find the value of pp

(1 mark)

The random variable TT represents the total number of seeds in the 33 packets.
(c) Find the sampling distribution of TT

(4 marks)
●●●●●Level 37 marksStart
A random variable XX has probability distribution defined as follows.

P(X=x)={k(x2+2)x=0,1,2,3,0otherwise,\mathrm{P}(X = x) = \begin{cases} k(x^2 + 2) & x = 0, 1, 2, 3, \\ 0 & \text{otherwise,} \end{cases}

where kk is a constant.
(a) Show that P(X=2)=311\mathrm{P}(X = 2) = \dfrac{3}{11}.

(3 marks)
(b) Show in a table the values of XX and their probabilities.

(2 marks)
●●●●●Level 35 marksStart
The discrete random variable XX has the probability distribution shown in the table, where kk is a constant.

x1234P(X=x)0.10.3k0.2\begin{array}{|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(X = x) & 0.1 & 0.3 & k & 0.2 \\ \hline\end{array}
(a) Find the value of kk. (1 mark)
(b) Find E(X)\mathrm{E}(X). (2 marks)
(c) Find Var(X)\mathrm{Var}(X). (3 marks)
(d) Find P(X>E(X))\mathrm{P}\big(X > \mathrm{E}(X)\big). (1 mark)
●●●●●Level 37 marksStart
The discrete random variable WW has cumulative distribution function F(w)\mathrm{F}(w) given by

F(w)=w(w+1)12w=1,2,3\mathrm{F}(w) = \frac{w(w + 1)}{12} \qquad w = 1, 2, 3

Find E(W)\mathrm{E}(W)

(4 marks)
●●●●●Level 34 marksStart

The binomial distribution as a model

32 questions

LessonNot started
A company that makes phone screens states that 96.5%96.5\% of its screens have no dead pixels. After a change of supplier, a random sample of 200200 screens is checked, and 22 of them are found to have a dead pixel.
(a) Explain why a Poisson distribution can be used to approximate the distribution of the number of screens in the sample that have a dead pixel.

(1 mark)
(b) Using a suitable approximation, test, at the 5%5\% level of significance, whether or not there is evidence that the proportion of screens with a dead pixel has decreased. State your hypotheses clearly.

(5 marks)
●●●●●Level 36 marksStart
Five cards are drawn without replacement from a standard pack, and XX is the number of hearts. Explain why XX does not follow a binomial distribution.
●●●●●Level 31 markStart
On average, 12%12\% of the tulip bulbs sold by a garden centre fail to flower.

A customer buys a bag of 2525 of these bulbs.

The number of bulbs in the bag that fail to flower can be modelled by the distribution B(25,0.12)\text{B}(25, 0.12)
(a) State one assumption, in context, required for this distribution to be used.

(1 mark)
(b) Find the probability that exactly 33 of the bulbs in the bag fail to flower.

(1 mark)
(c) Find the probability that fewer than 55 of the bulbs in the bag fail to flower.

(2 marks)
●●●●●Level 34 marksStart
Twenty seeds are planted; each germinates independently with probability 0.80.8. Suggest a suitable model for the number that germinate, and find the probability that all 2020 germinate, to 4 d.p.
●●●●●Level 33 marksStart
A multiple-choice test has 1515 questions, each with 44 options. A student guesses every answer. Let XX be the number correct.
(a) State the distribution of XX.
(b) Find P(X=5)P(X = 5), to 4 d.p.
●●●●●Level 33 marksStart
At a market stall, customers are offered a taste of a cheese before they decide whether to buy any.

Over a long period, Lena has found that 40%40\% of the customers who taste this cheese then buy some.

Lena changes the way the cheese is displayed on her stall and claims that the proportion of customers who taste the cheese and then buy some has increased.

A random sample of 3030 customers who taste the cheese after the change is taken and 1818 of them buy some cheese.

Use a suitable test to assess Lena's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
●●●●●Level 34 marksStart
Nadia is practising on a climbing wall.

Each time Nadia attempts a particular route, the probability that she completes the route without resting is 0.350.35

Nadia attempts the route 1818 times.

The random variable XX represents the number of times Nadia completes the route without resting.
(a) State a condition needed for XX to be modelled by a binomial distribution.

(1 mark)
(b) Using a suitable distribution for XX, find

(i) P(X=5)\mathrm{P}(X = 5)

(ii) P(X⩾8)\mathrm{P}(X \geqslant 8)

(3 marks)
●●●●●Level 34 marksStart
A café runs a promotion in which every hot drink sold comes with a scratch card.

For each scratch card
  • the card either wins a free pastry or does not win a free pastry
  • cards win independently of each other
The probability that a scratch card wins a free pastry is 0.150.15
(a) Find the probability that

(i) exactly 33 of 1212 scratch cards win a free pastry,

(ii) at least 44 of 3030 scratch cards win a free pastry.

(4 marks)

Maya buys one hot drink from the café on each of nn days.
(b) Find the minimum value of nn such that the probability that at least one of Maya's scratch cards wins a free pastry exceeds 0.90.9

(4 marks)
●●●●●Level 38 marksStart

Calculating binomial probabilities P(X = r)

40 questions

LessonNot started
X∼B(20,0.15)X \sim B(20, 0.15). Find P(X=2)P(X = 2), to 4 d.p.
●●●●●Level 32 marksStart
X∼B(10,0.7)X \sim B(10, 0.7). Find P(X=8)P(X = 8), to 4 d.p.
●●●●●Level 32 marksStart
On any given day, the probability that a certain machine breaks down is 0.150.15, independently of all other days.

Find the probability that, in a period of 1212 days, the machine breaks down
(a) exactly 33 times, (2 marks)
(b) at least 33 times. (3 marks)
●●●●●Level 35 marksStart
A fair coin is tossed 88 times. Find the probability of exactly 44 heads, to 4 d.p.
●●●●●Level 32 marksStart
X∼B(25,0.2)X \sim B(25, 0.2). Find P(X=5)P(X = 5), to 4 d.p.
●●●●●Level 32 marksStart
A spinner lands on red with probability 0.30.3. It is spun 1010 times. Find the probability of exactly 44 reds, to 4 d.p.
●●●●●Level 32 marksStart
On average, 12%12\% of the tulip bulbs sold by a garden centre fail to flower.

A customer buys a bag of 2525 of these bulbs.

The number of bulbs in the bag that fail to flower can be modelled by the distribution B(25,0.12)\text{B}(25, 0.12)
(a) State one assumption, in context, required for this distribution to be used.

(1 mark)
(b) Find the probability that exactly 33 of the bulbs in the bag fail to flower.

(1 mark)
(c) Find the probability that fewer than 55 of the bulbs in the bag fail to flower.

(2 marks)
●●●●●Level 34 marksStart
X∼B ⁣(6,13)X \sim B\!\left(6, \tfrac13\right).
(a) Find P(X=2)P(X = 2).
(b) Find P(X=0)P(X = 0). Give answers to 4 d.p.
●●●●●Level 34 marksStart
Nadia is practising on a climbing wall.

Each time Nadia attempts a particular route, the probability that she completes the route without resting is 0.350.35

Nadia attempts the route 1818 times.

The random variable XX represents the number of times Nadia completes the route without resting.
(a) State a condition needed for XX to be modelled by a binomial distribution.

(1 mark)
(b) Using a suitable distribution for XX, find

(i) P(X=5)\mathrm{P}(X = 5)

(ii) P(X⩾8)\mathrm{P}(X \geqslant 8)

(3 marks)
●●●●●Level 34 marksStart
A café runs a promotion in which every hot drink sold comes with a scratch card.

For each scratch card
  • the card either wins a free pastry or does not win a free pastry
  • cards win independently of each other
The probability that a scratch card wins a free pastry is 0.150.15
(a) Find the probability that

(i) exactly 33 of 1212 scratch cards win a free pastry,

(ii) at least 44 of 3030 scratch cards win a free pastry.

(4 marks)

Maya buys one hot drink from the café on each of nn days.
(b) Find the minimum value of nn such that the probability that at least one of Maya's scratch cards wins a free pastry exceeds 0.90.9

(4 marks)
●●●●●Level 38 marksStart

Cumulative binomial probabilities

39 questions

LessonNot started
X∼B(30,0.5)X \sim B(30, 0.5). Find P(12≤X≤18)P(12 \le X \le 18), to 4 d.p.
●●●●●Level 32 marksStart
X∼B(20,0.3)X \sim B(20, 0.3). Find P(X≤5)P(X \le 5), to 4 d.p.
●●●●●Level 32 marksStart
X∼B(25,0.4)X \sim B(25, 0.4). Find P(X<10)P(X < 10), to 4 d.p.
●●●●●Level 32 marksStart
On any given day, the probability that a certain machine breaks down is 0.150.15, independently of all other days.

Find the probability that, in a period of 1212 days, the machine breaks down
(a) exactly 33 times, (2 marks)
(b) at least 33 times. (3 marks)
●●●●●Level 35 marksStart
12%12\% of components from a line are defective. In a sample of 5050, find the probability that at most 44 are defective, to 4 d.p.
●●●●●Level 32 marksStart
On average, 12%12\% of the tulip bulbs sold by a garden centre fail to flower.

A customer buys a bag of 2525 of these bulbs.

The number of bulbs in the bag that fail to flower can be modelled by the distribution B(25,0.12)\text{B}(25, 0.12)
(a) State one assumption, in context, required for this distribution to be used.

(1 mark)
(b) Find the probability that exactly 33 of the bulbs in the bag fail to flower.

(1 mark)
(c) Find the probability that fewer than 55 of the bulbs in the bag fail to flower.

(2 marks)
●●●●●Level 34 marksStart
X∼B(20,0.25)X \sim B(20, 0.25). Find P(X>7)P(X > 7), to 4 d.p.
●●●●●Level 32 marksStart
At a market stall, customers are offered a taste of a cheese before they decide whether to buy any.

Over a long period, Lena has found that 40%40\% of the customers who taste this cheese then buy some.

Lena changes the way the cheese is displayed on her stall and claims that the proportion of customers who taste the cheese and then buy some has increased.

A random sample of 3030 customers who taste the cheese after the change is taken and 1818 of them buy some cheese.

Use a suitable test to assess Lena's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
●●●●●Level 34 marksStart
Nadia is practising on a climbing wall.

Each time Nadia attempts a particular route, the probability that she completes the route without resting is 0.350.35

Nadia attempts the route 1818 times.

The random variable XX represents the number of times Nadia completes the route without resting.
(a) State a condition needed for XX to be modelled by a binomial distribution.

(1 mark)
(b) Using a suitable distribution for XX, find

(i) P(X=5)\mathrm{P}(X = 5)

(ii) P(X⩾8)\mathrm{P}(X \geqslant 8)

(3 marks)
●●●●●Level 34 marksStart
X∼B(15,0.2)X \sim B(15, 0.2). Find P(X≥4)P(X \ge 4), to 4 d.p.
●●●●●Level 32 marksStart
X∼B(18,0.35)X \sim B(18, 0.35). Find P(5≤X≤9)P(5 \le X \le 9), to 4 d.p.
●●●●●Level 32 marksStart
X∼B(40,0.1)X \sim B(40, 0.1). Find P(X≥3)P(X \ge 3), to 4 d.p.
●●●●●Level 32 marksStart
A café runs a promotion in which every hot drink sold comes with a scratch card.

For each scratch card
  • the card either wins a free pastry or does not win a free pastry
  • cards win independently of each other
The probability that a scratch card wins a free pastry is 0.150.15
(a) Find the probability that

(i) exactly 33 of 1212 scratch cards win a free pastry,

(ii) at least 44 of 3030 scratch cards win a free pastry.

(4 marks)

Maya buys one hot drink from the café on each of nn days.
(b) Find the minimum value of nn such that the probability that at least one of Maya's scratch cards wins a free pastry exceeds 0.90.9

(4 marks)
●●●●●Level 38 marksStart

The normal distribution

38 questions

LessonNot started
X∼N(50,σ2)X \sim N(50, \sigma^2) and P(X<60)=0.84P(X < 60) = 0.84. Find P(X>60)P(X > 60) and P(40<X<60)P(40 < X < 60).
●●●●●Level 33 marksStart
X∼N(40,9)X \sim N(40, 9). Find P(X<43)P(X < 43), giving your answer to 4 decimal places.
●●●●●Level 32 marksStart
X∼N(100,225)X \sim N(100, 225). Find the zz-scores of X=85X = 85 and X=130X = 130.
●●●●●Level 32 marksStart
X∼N(30,4)X \sim N(30, 4). State the values of XX at which the curve has its points of inflection.
●●●●●Level 32 marksStart
A student scores 7575 in a maths test N(60,100)N(60, 100) and 6666 in an English test N(50,64)N(50, 64). By comparing zz-scores, state in which subject the student did relatively better.
●●●●●Level 33 marksStart
A grower measures the cucumbers picked from a crop in a greenhouse.

The random variable L∼N(μ,σ2)L \sim \mathrm{N}(\mu, \sigma^2) represents the length, in cm, of a cucumber picked from this crop.

Given that P(L<30)=0.15\mathrm{P}(L < 30) = 0.15 and P(L<45)=0.90\mathrm{P}(L < 45) = 0.90

calculate the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 35 marksStart
X∼N(60,64)X \sim N(60, 64). Find the zz-score of the value X=72X = 72.
●●●●●Level 32 marksStart
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The volume, VV ml, of coffee dispensed into a cup by a vending machine can be modelled by a normal distribution with mean 236 ml and standard deviation 6 ml.
(a) Find the probability that a randomly selected cup contains at least 248 ml of coffee.

(2 marks)

The probability that a randomly selected cup contains at least dd ml of coffee is 0.95
(b) Find the value of dd

(3 marks)
●●●●●Level 35 marksStart
The variable XX has the distribution N(48,25)\mathrm{N}(48, 25).
(a) Find P(X<41)\mathrm{P}(X < 41).

(1 mark)
(b) Find bb such that P(48−b<X<48+b)=0.7\mathrm{P}(48 - b < X < 48 + b) = 0.7.

(3 marks)
●●●●●Level 34 marksStart
The random variable X∼N(50,16)X \sim N(50, 16).
(a) Find P(X>56)P(X > 56).
(b) Find P(46<X<56)P(46 < X < 56).

Give each answer correct to 4 decimal places.
●●●●●Level 35 marksStart

Finding normal probabilities

51 questions

LessonNot started
X∼N(80,36)X \sim N(80, 36).

Find the value exceeded by the top 5%5\% of the distribution, giving your answer to 3 significant figures. (3 marks)
●●●●●Level 33 marksStart
IQ scores in a population are modelled by X∼N(100,152)X \sim N(100, 15^{2}).
(a) Find P(X>130)P(X > 130), giving your answer to 3 significant figures. (2 marks)
(b) In a town of 10001000 people, estimate how many have an IQ above 130130. (2 marks)
●●●●●Level 34 marksStart
X∼N(70,25)X \sim N(70, 25). Find the value of kk for which P(70−k<X<70+k)=0.95P(70 - k < X < 70 + k) = 0.95, to 3 s.f.
●●●●●Level 32 marksStart
X∼N(20,25)X \sim N(20, 25). Find P(X<28)P(X < 28), to 4 d.p.
●●●●●Level 32 marksStart
X∼N(30,25)X \sim N(30, 25).

Find P(24<X<36)P(24 < X < 36), giving your answer to 4 decimal places. (3 marks)
●●●●●Level 33 marksStart
X∼N(60,16)X \sim N(60, 16).

Find the value of aa such that P(X<a)=0.95P(X < a) = 0.95, giving your answer to 3 significant figures. (3 marks)
●●●●●Level 33 marksStart
A grower measures the cucumbers picked from a crop in a greenhouse.

The random variable L∼N(μ,σ2)L \sim \mathrm{N}(\mu, \sigma^2) represents the length, in cm, of a cucumber picked from this crop.

Given that P(L<30)=0.15\mathrm{P}(L < 30) = 0.15 and P(L<45)=0.90\mathrm{P}(L < 45) = 0.90

calculate the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 35 marksStart
X∼N(100,225)X \sim N(100, 225). Find the value aa such that P(X<a)=0.90P(X < a) = 0.90, to 3 s.f.
●●●●●Level 33 marksStart
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The volume, VV ml, of coffee dispensed into a cup by a vending machine can be modelled by a normal distribution with mean 236 ml and standard deviation 6 ml.
(a) Find the probability that a randomly selected cup contains at least 248 ml of coffee.

(2 marks)

The probability that a randomly selected cup contains at least dd ml of coffee is 0.95
(b) Find the value of dd

(3 marks)
●●●●●Level 35 marksStart
The variable XX has the distribution N(48,25)\mathrm{N}(48, 25).
(a) Find P(X<41)\mathrm{P}(X < 41).

(1 mark)
(b) Find bb such that P(48−b<X<48+b)=0.7\mathrm{P}(48 - b < X < 48 + b) = 0.7.

(3 marks)
●●●●●Level 34 marksStart
X∼N(50,16)X \sim N(50, 16). Find P(46<X<56)P(46 < X < 56), to 4 d.p.
●●●●●Level 32 marksStart
Scores are modelled by X∼N(500,1600)X \sim N(500, 1600). Find the score exceeded by the top 5%5\% of candidates, to 3 s.f.
●●●●●Level 33 marksStart
X∼N(50,100)X \sim N(50, 100).

Find P(X>60∣X>55)P(X > 60 \mid X > 55), giving your answer to 3 significant figures. (5 marks)
●●●●●Level 35 marksStart

Finding μ and σ

33 questions

LessonNot started
X∼N(μ,σ2)X \sim N(\mu, \sigma^{2}) with P(X<20)=0.2P(X < 20) = 0.2 and P(X>60)=0.2P(X > 60) = 0.2.
(a) Write down the value of μ\mu, justifying your answer. (2 marks)
(b) Find σ\sigma, giving your answer to 3 significant figures. (2 marks)
●●●●●Level 34 marksStart
X∼N(100,σ2)X \sim N(100, \sigma^{2}) and P(95<X<105)=0.80P(95 < X < 105) = 0.80.

Find σ\sigma, giving your answer to 3 significant figures. (4 marks)
●●●●●Level 34 marksStart
The masses of loaves from a bakery are modelled as N(μ,42)N(\mu, 4^{2}) grams. Exactly 3%3\% of loaves weigh less than 500500 g.

Find μ\mu, giving your answer to 3 significant figures. (3 marks)
●●●●●Level 33 marksStart
X∼N(100,σ2)X \sim N(100, \sigma^2) and P(X<115)=0.95P(X < 115) = 0.95. Find σ\sigma, to 3 s.f.
●●●●●Level 33 marksStart
X∼N(μ,25)X \sim N(\mu, 25) and P(X>30)=0.025P(X > 30) = 0.025. Find μ\mu.
●●●●●Level 33 marksStart
X∼N(50,σ2)X \sim N(50, \sigma^2) and P(X>60)=0.20P(X > 60) = 0.20. Find σ\sigma, to 3 s.f.
●●●●●Level 33 marksStart
Bottle volumes are N(μ,σ2)N(\mu, \sigma^2) ml, with P(X<1000)=0.10P(X < 1000) = 0.10 and P(X>1050)=0.05P(X > 1050) = 0.05. Find μ\mu and σ\sigma, to 3 s.f.
●●●●●Level 34 marksStart
X∼N(μ,9)X \sim N(\mu, 9) and P(X<50)=0.975P(X < 50) = 0.975.

Find μ\mu, giving your answer to 3 significant figures. (3 marks)
●●●●●Level 33 marksStart
X∼N(40,σ2)X \sim N(40, \sigma^{2}) and P(X<49)=0.85P(X < 49) = 0.85.

Find σ\sigma, giving your answer to 3 significant figures. (3 marks)
●●●●●Level 33 marksStart
A grower measures the cucumbers picked from a crop in a greenhouse.

The random variable L∼N(μ,σ2)L \sim \mathrm{N}(\mu, \sigma^2) represents the length, in cm, of a cucumber picked from this crop.

Given that P(L<30)=0.15\mathrm{P}(L < 30) = 0.15 and P(L<45)=0.90\mathrm{P}(L < 45) = 0.90

calculate the value of μ\mu and the value of σ\sigma

(5 marks)
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X∼N(μ,16)X \sim N(\mu, 16) and P(X<30)=0.90P(X < 30) = 0.90. Find μ\mu, to 3 s.f.
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X∼N(μ,σ2)X \sim N(\mu, \sigma^2) with P(X<20)=0.10P(X < 20) = 0.10 and P(X<40)=0.75P(X < 40) = 0.75. Find μ\mu and σ\sigma, to 3 s.f.
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X∼N(μ,σ2)X \sim N(\mu, \sigma^2) with P(X>50)=0.30P(X > 50) = 0.30 and P(X>20)=0.80P(X > 20) = 0.80. Find μ\mu and σ\sigma, to 3 s.f.
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Approximating a binomial with a normal

24 questions

LessonNot started
X∼B(500,0.2)X \sim B(500, 0.2).

Use a normal approximation to estimate P(90⩽X⩽110)P(90 \leqslant X \leqslant 110), giving your answer to 3 significant figures. (5 marks)
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X∼B(200,0.3)X \sim B(200, 0.3). Use a normal approximation to estimate P(X≥70)P(X \ge 70), to 3 d.p.
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X∼B(400,0.5)X \sim B(400, 0.5).

Use a normal approximation to estimate P(X⩽210)P(X \leqslant 210), giving your answer to 3 significant figures. (4 marks)
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X∼B(80,0.4)X \sim B(80, 0.4). State the normal distribution that approximates XX.
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A fair coin is flipped 6464 times.

Use a normal approximation to estimate the probability of getting at least 4040 heads, giving your answer to 3 significant figures. (4 marks)
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X∼B(120,0.25)X \sim B(120, 0.25). Use a normal approximation to estimate P(25≤X≤35)P(25 \le X \le 35), to 3 d.p.
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X∼B(100,0.5)X \sim B(100, 0.5). Use a normal approximation to estimate P(X>55)P(X > 55), to 3 d.p.
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X∼B(60,0.5)X \sim B(60, 0.5). Use a normal approximation to estimate P(X≤30)P(X \le 30), to 3 d.p.
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X∼B(300,0.6)X \sim B(300, 0.6).
(a) State the normal distribution that approximates XX. (2 marks)
(b) Give one reason why the approximation is appropriate here. (1 mark)
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X∼B(180,0.5)X \sim B(180, 0.5).

Use a normal approximation to estimate P(X⩾100)P(X \geqslant 100), giving your answer to 3 significant figures. (4 marks)
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X∼B(100,0.6)X \sim B(100, 0.6).

Use a normal approximation to estimate P(X=60)P(X = 60), giving your answer to 3 significant figures. (4 marks)
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X∼B(100,0.5)X \sim B(100, 0.5). Use a normal approximation to estimate P(X<45)P(X < 45), to 3 d.p.
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X∼B(50,0.5)X \sim B(50, 0.5). Use a normal approximation to estimate P(X=25)P(X = 25), to 3 d.p.
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