Direct binomial from normal probability

2 The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm .

1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm .
2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm .
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   The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies.

6 The lengths, in centimetres, of drinking straws produced in a factory have a normal distribution with mean \(\mu\) and variance 0.64 . It is given that \(10 \%\) of the straws are shorter than 20 cm .

1. Find the value of \(\mu\).
2. Find the probability that, of 4 straws chosen at random, fewer than 2 will have a length between 21.5 cm and 22.5 cm .

5 When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.

1. \(90 \%\) of Moses's phone calls take longer than \(t\) minutes. Find the value of \(t\).
2. Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( - 3 , \sigma ^ { 2 } \right)\). The probability that a randomly chosen value of \(X\) is positive is 0.25 .

1. Find the value of \(\sigma\).
2. Find the probability that, of 8 random values of \(X\), fewer than 2 will be positive.

2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .

5 The weights of apples sold by a store can be modelled by a normal distribution with mean 120 grams and standard deviation 24 grams. Apples weighing less than 90 grams are graded as 'small'; apples weighing more than 140 grams are graded as 'large'; the remainder are graded as 'medium'.

1. Show that the probability that an apple chosen at random is graded as medium is 0.692 , correct to 3 significant figures.
2. Four apples are chosen at random. Find the probability that at least two are graded as medium. [4]

2 The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7 cm and standard deviation 1.6 cm .

1. Find the probability that the head circumference of a randomly selected 3 -year-old boy will be
   (A) over 51.5 cm ,
   (B) between 48.0 and 51.5 cm .
2. Four 3-year-old boys are selected at random. Find the probability that exactly one of them has head circumference between 48.0 and 51.5 cm .
3. The head circumference of 3-year-old girls is known to be Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(60 \%\) of 3-year-old girls have head circumference below 49.0 cm and \(30 \%\) have head circumference below 47.5 cm , find the values of \(\mu\) and \(\sigma\). A nutritionist claims that boys who have been fed on a special organic diet will have a larger mean head circumference than other boys. A random sample of ten 3 -year-old boys who have been fed on this organic diet is selected. It is found that their mean head circumference is 50.45 cm .
4. Using the null and alternative hypotheses \(\mathrm { H } _ { 0 } : \mu = 49.7 \mathrm {~cm} , \mathrm { H } _ { 1 } : \mu > 49.7 \mathrm {~cm}\), carry out a test at the \(10 \%\) significance level to examine the nutritionist's claim. Explain the meaning of \(\mu\) in these hypotheses. You may assume that the standard deviation of the head circumference of organically fed 3 -year-old boys is 1.6 cm .

3 A company has a fleet of identical vans. Company policy is to replace all of the tyres on a van as soon as any one of them is worn out. The random variable \(X\) represents the number of miles driven before the tyres on a van are replaced. \(X\) is Normally distributed with mean 27500 and standard deviation 4000.

1. Find \(\mathrm { P } ( X > 25000 )\).
2. 10 vans in the fleet are selected at random. Find the probability that the tyres on exactly 7 of them last for more than 25000 miles.
3. The tyres of \(99 \%\) of vans last for more than \(k\) miles. Find the value of \(k\). A tyre supplier claims that a different type of tyre will have a greater mean lifetime. A random sample of 15 vans is fitted with these tyres. For each van, the number of miles driven before the tyres are replaced is recorded. A hypothesis test is carried out to investigate the claim. You may assume that these lifetimes are also Normally distributed with standard deviation 4000.
4. Write down suitable null and alternative hypotheses for the test.
5. For the 15 vans, it is found that the mean lifetime of the tyres is 28630 miles. Carry out the test at the \(5 \%\) level.

3 The lifetime of a particular type of light bulb is \(X\) hours, where \(X\) is Normally distributed with mean 1100 and variance 2000.

1. Find \(\mathrm { P } ( 1100 < X < 1200 )\).
2. Use a suitable approximating distribution to find the probability that, in a random sample of 100 of these light bulbs, no more than 40 have a lifetime between 1100 and 1200 hours.
3. A factory has a large number of these light bulbs installed. As soon as \(1 \%\) of the bulbs have come to the end of their lifetimes, it is company policy to replace all of the bulbs. After how many hours should the bulbs need to be replaced?
4. The bulbs are to be replaced by low-energy bulbs. The lifetime of these bulbs is Normally distributed and the mean is claimed by the manufacturer to be 7000 hours. The standard deviation is known to be 100 hours. A random sample of 25 low-energy bulbs is selected. Their mean lifetime is found to be 6972 hours. Carry out a 2 -tail test at the \(10 \%\) level to investigate the claim.
   [0pt] [Question 4 is printed overleaf.]

3 The random variable \(X\) represents the weight in kg of a randomly selected male dog of a particular breed. \(X\) is Normally distributed with mean 30.7 and standard deviation 3.5.

1. Find
   (A) \(\mathrm { P } ( X < 30 )\),
   (B) \(P ( 25 < X < 35 )\).
2. Five of these dogs are chosen at random. Find the probability that each of them weighs at least 30 kg .
3. The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg . Given that \(5 \%\) of female dogs of this breed weigh more than 30 kg , find the standard deviation of their weights.
4. Sketch the distributions of the weights of male and female dogs of this breed on a single diagram.

9

1. The masses, in grams, of plums of a certain kind have the distribution \(\mathrm { N } ( 55,18 )\).
   1. Find the probability that a plum chosen at random has a mass between 50.0 and 60.0 grams.
   2. The heaviest \(5 \%\) of plums are classified as extra large. Find the minimum mass of extra large plums.
   3. The plums are packed in bags, each containing 10 randomly selected plums. Find the probability that a bag chosen at random has a total mass of less than 530 g .
2. The masses, in grams, of apples of a certain kind have the distribution \(\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)\). It is given that half of the apples have masses between 62 g and 72 g . Determine \(\sigma\).

1. A machine squeezes apples to extract their juice. The volume of juice, \(J \mathrm { ml }\), extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\)

Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to

1. 1. show that \(\mathrm { P } ( J > 510 ) = 0.3446\)
   2. calculate the value of \(d\) such that \(\mathrm { P } ( J > d ) = 0.9192\) Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
2. Calculate the probability that each of the 5 bags of apples produce less than 510 ml of juice. Following adjustments to the machine, the volume of juice, \(R \mathrm { ml }\), extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that \(\mathrm { P } ( R < r ) = 0.15\) and \(\mathrm { P } ( R > 3 r - 800 ) = 0.005\)
3. find the value of \(r\) and the value of \(k\)

1. A machine makes bolts such that the length, \(L \mathrm {~cm}\), of a bolt has distribution \(L \sim \mathrm {~N} \left( 4.1,0.125 ^ { 2 } \right)\)

A bolt is selected at random.

1. Find the probability that the length of this bolt is more than 4.3 cm .
2. Show that \(\mathrm { P } ( 3.9 < L < 4.3 )\) is 0.890 correct to 3 decimal places. The machine makes 500 bolts.
   The cost to make each bolt is 5 pence.
   Only bolts with length between 3.9 cm and 4.3 cm can be used. These are sold for 9 pence each. All the bolts that cannot be used are recycled with a scrap value of 1 pence each.
3. Calculate an estimate for the profit made on these 500 bolts. Following adjustments to the machine, the length of a bolt, \(B \mathrm {~cm}\), made by the machine is such that \(B \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( B > 4.198 ) = 0.025\) and \(\mathrm { P } ( B < 4.065 ) = 0.242\)
4. find the value of \(\mu\) and the value of \(\sigma\)
5. State, giving a reason, whether the adjustments to the machine will result in a decrease or an increase in the profit made on 500 bolts.

3. The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.
i. Find the probability that a randomly selected apple weighs at least 220 grams.
ii. 80 apples are selected at random.
a) Find the probability that more than 18 of these apples weigh at least 220 grams.
b) Find the expectation and standard deviation for the number of apples that weigh at least 220 grams.
c) State a suitable approximating distribution, including any parameters, for the number of apples that weigh at least 220 grams.
d) Explain why this approximating distribution is suitable. The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation \(\sigma\).
iii. Given that \(10 \%\) of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of \(\sigma\).