Direct binomial from normal probability

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.

7 The weights, \(X\) grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.

1. Find the probability that a randomly chosen bar of soap weighs more than 128 grams.
2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 128 ) = 0.7465\).
3. Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than 128 grams.

2 The random variable \(X\) is the daily profit, in thousands of dollars, made by a company. \(X\) is normally distributed with mean 6.4 and standard deviation 5.2.

1. Find the probability that, on a randomly chosen day, the company makes a profit between \(\\) 10000\( and \)\\( 12000\).
2. Find the probability that the company makes a loss on exactly 1 of the next 4 consecutive days.

5 Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm .

1. \(8 \%\) of carrots are shorter than \(c \mathrm {~cm}\). Find the value of \(c\).
2. Rebekah picks 7 carrots at random. Find the probability that at least 2 of them have lengths between 15 and 16 cm .

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]

7 The shortest time recorded by an athlete in a 400 m race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean 49.2 seconds and standard deviation 2.8 seconds.

1. Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds.
2. It is found that \(92 \%\) of athletes from this club have PBs of more than \(t\) seconds. Find the value of \(t\).
   Three athletes from the club are chosen at random.
3. Find the probability that exactly 2 have PBs of less than 46 seconds.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 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 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.

7. A packing plant fills bags with cement. The weight \(X \mathrm {~kg}\) of a bag of cement can be modelled by a normal distribution with mean 50 kg and standard deviation 2 kg .

1. Find \(\mathrm { P } ( X > 53 )\).
2. Find the weight that is exceeded by \(99 \%\) of the bags. Three bags are selected at random.
3. Find the probability that two weigh more than 53 kg and one weighs less than 53 kg .

4 A tomato grower grows just one variety of tomatoes. The weights of these tomatoes are found to be normally distributed with a mean of 85.1 grams and a standard deviation of 3.4 grams.

1. Find the probability that a randomly chosen tomato of this variety weighs less than 80 grams.
2. The grower puts the tomatoes in packs of 6 . Find the probability that, in a randomly chosen pack of 6 , at most one tomato weighs less than 80 grams.
3. The grower supplies consignments of 250 packs of these tomatoes to a retailer. For a randomly chosen consignment, find the expected number of packs having more than one tomato weighing less than 80 grams.

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]
2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm. [3]