2.04e Normal distribution: as model N(mu, sigma^2)

7 The continuous random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The mean of a random sample of \(n\) observations of \(X\) is denoted by \(\bar { X }\). It is given that \(\mathrm { P } ( \bar { X } < 35.0 ) = 0.9772\) and \(\mathrm { P } ( \bar { X } < 20.0 ) = 0.1587\).

1. Obtain a formula for \(\sigma\) in terms of \(n\). Two students are discussing this question. Aidan says "If you were told another probability, for instance \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\), you could work out the value of \(\sigma\)." Binya says, "No, the value of \(\mathrm { P } ( \bar { X } > 32 )\) is fixed by the information you know already."
2. State which of Aidan and Binya is right. If you think that Aidan is right, calculate the value of \(\sigma\) given that \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\). If you think that Binya is right, calculate the value of \(\mathrm { P } ( \bar { X } > 32 )\).

1 The random variable \(Y\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is found that \(\mathrm { P } ( Y > 150.0 ) = 0.0228\) and \(\mathrm { P } ( Y > 143.0 ) = 0.9332\). Find the values of \(\mu\) and \(\sigma\).

4 A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.

1. Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
   The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
2. Find the value of \(m\).
3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
   

2 The continuous random variable \(Y\) has the distribution \(\mathrm { N } \left( 23.0,5.0 ^ { 2 } \right)\). The mean of \(n\) observations of \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } > 23.625 ) = 0.0228\). Find the value of \(n\).

6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).

1. Given that \(n = 81\),
   1. find the critical region for the test, in terms of the sample mean \(\bar { W }\),
   2. find the probability that the test results in a Type II error when \(\mu = 50.2\).
   3. State how the probability of this Type II error would change if \(n\) were greater than 81 .

2 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). The mean of a sample of \(n\) observations of \(H\) is denoted by \(\bar { H }\). It is given that \(\mathrm { P } ( \bar { H } > 53.28 ) = 0.0250\) and \(\mathrm { P } ( \bar { H } < 51.65 ) = 0.0968\), both correct to 4 decimal places. Find the values of \(\mu\) and \(n\).

1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 105.0 ) = 0.2420\) and \(\mathrm { P } ( H > 110.0 ) = 0.6915\). Find the values of \(\mu\) and \(\sigma\), giving your answers to a suitable degree of accuracy.

6 The continuous random variable \(R\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 100 observations of \(R\) are summarised by $$\Sigma r = 3360.0 , \quad \Sigma r ^ { 2 } = 115782.84 .$$

1. Calculate an unbiased estimate of \(\mu\) and an unbiased estimate of \(\sigma ^ { 2 }\).
2. The mean of 9 observations of \(R\) is denoted by \(\bar { R }\). Calculate an estimate of \(\mathrm { P } ( \bar { R } > 32.0 )\).
3. Explain whether you need to use the Central Limit Theorem in your answer to part (ii).

3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).

1. Find the value of \(\sigma\).
2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).

2 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that $$\mathrm { P } ( Y < 48.0 ) = \mathrm { P } ( Y > 57.0 ) = 0.0668 .$$ Find the value \(y _ { 0 }\) such that \(\mathrm { P } \left( Y > y _ { 0 } \right) = 0.05\).

3 The number of minutes, \(X\), for which a particular model of laptop computer will run on battery power is Normally distributed with mean 115.3 and standard deviation 21.9.

1. (A) Find \(\mathrm { P } ( X < 120 )\).
   (B) Find \(\mathrm { P } ( 100 < X < 110 )\).
   (C) Find the value of \(k\) for which \(\mathrm { P } ( X > k ) = 0.9\). The number of minutes, \(Y\), for which a different model of laptop computer will run on battery power is known to be Normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
2. Given that \(\mathrm { P } ( Y < 180 ) = 0.7\) and \(\mathrm { P } ( Y < 140 ) = 0.15\), find the values of \(\mu\) and \(\sigma\).
3. Find values of \(a\) and \(b\) for which \(\mathrm { P } ( a < Y < b ) = 0.95\).

3 In an English language test for 12-year-old children, the raw scores, \(X\), are Normally distributed with mean 45.3 and standard deviation 11.5.

1. Find
   (A) \(\mathrm { P } ( X < 50 )\),
   (B) \(\mathrm { P } ( 45.3 < X < 50 )\).
2. Find the least raw score which would be obtained by the highest scoring \(10 \%\) of children.
3. The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
4. In a Mathematics test for 12-year-old children, the raw scores, \(Y\), are Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\mathrm { P } ( Y < 15 ) = 0.3\) and \(\mathrm { P } ( Y < 22 ) = 0.8\), find the values of \(\mu\) and \(\sigma\).

3 The random variable \(X\) represents the reaction times, in milliseconds, of men in a driving simulator. \(X\) is Normally distributed with mean 355 and standard deviation 52.

1. Find
   (A) \(\mathrm { P } ( X < 325 )\),
   (B) \(\mathrm { P } ( 300 < X < 400 )\).
2. Find the value of \(k\) for which \(\mathrm { P } ( X < k ) = 0.2\). It is thought that women may have a different mean reaction time from men. In order to test this, a random sample of 25 women is selected. The mean reaction time of these women in the driving simulator is 344 milliseconds. You may assume that women's reaction times are also Normally distributed with standard deviation 52 milliseconds. A hypothesis test is carried out to investigate whether women have a different mean reaction time from men.
3. Carry out the test at the \(5 \%\) significance 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 amount of data, \(X\) megabytes, arriving at an internet server per second during the afternoon is modelled by the Normal distribution with mean 435 and standard deviation 30.

1. Find
   (A) \(\mathrm { P } ( X < 450 )\),
   (B) \(\mathrm { P } ( 400 < X < 450 )\).
2. Find the probability that, during 5 randomly selected seconds, the amounts of data arriving are all between 400 and 450 megabytes. The amount of data, \(Y\) megabytes, arriving at the server during the evening is modelled by the Normal distribution with mean \(\mu\) and standard deviation \(\sigma\).
3. Given that \(\mathrm { P } ( Y < 350 ) = 0.2\) and \(\mathrm { P } ( Y > 390 ) = 0.1\), find the values of \(\mu\) and \(\sigma\).
4. Find values of \(a\) and \(b\) for which \(\mathrm { P } ( a < Y < b ) = 0.95\).

3 Intensity of light is measured in lumens. The random variable \(X\) represents the intensity of the light from a standard 100 watt light bulb. \(X\) is Normally distributed with mean 1720 and standard deviation 90. You may assume that the intensities for different bulbs are independent.

1. Show that \(\mathrm { P } ( X < 1700 ) = 0.4121\).
2. These bulbs are sold in packs of 4 . Find the probability that the intensities of exactly 2 of the 4 bulbs in a randomly chosen pack are below 1700 lumens.
3. Use a suitable approximating distribution to find the probability that the intensities of at least 20 out of 40 randomly selected bulbs are below 1700 lumens. A manufacturer claims that the average intensity of its 25 watt low energy light bulbs is 1720 lumens. A consumer organisation suspects that the true figure may be lower than this. The intensities of a random sample of 20 of these bulbs are measured. A hypothesis test is then carried out to check the claim.
4. Write down a suitable null hypothesis and explain briefly why the alternative hypothesis should be \(\mathrm { H } _ { 1 } : \mu < 1720\). State the meaning of \(\mu\).
5. Given that the standard deviation of the intensity of such bulbs is 90 lumens and that the mean intensity of the sample of 20 bulbs is 1703 lumens, carry out the test at the \(5 \%\) significance level.

3 In a men's cycling time trial, the times are modelled by the random variable \(X\) minutes which is Normally distributed with mean 63 and standard deviation 5.2.

1. Find $$\begin{aligned} & \text { (A) } \mathrm { P } ( X < 65 ) \text {, } \\ & \text { (B) } \mathrm { P } ( 60 < X < 65 ) \text {. } \end{aligned}$$
2. Find the probability that 5 riders selected at random all record times between 60 and 65 minutes.
3. A competitor aims to be in the fastest \(5 \%\) of entrants (i.e. those with the lowest times). Find the maximum time that he can take. It is suggested that holding the time trial on a new course may result in lower times. To investigate this, a random sample of 15 competitors is selected. These 15 competitors do the time trial on the new course. The mean time taken by these riders is 61.7 minutes. You may assume that times are Normally distributed and the standard deviation is still 5.2 minutes. A hypothesis test is carried out to investigate whether times on the new course are lower.
4. Write down suitable null and alternative hypotheses for the test. Carry out the test at the 5\% significance level.

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.

1. Find the probability that a randomly selected apple weighs at least 220 grams.
2. These apples are sold in packs of 3. You may assume that the weights of apples in each pack are independent. Find the probability that all 3 of the apples in a randomly selected pack weigh at least 220 grams.
3. 100 packs are selected at random.
   (A) State the exact distribution of the number of these 100 packs in which all 3 apples weigh at least 220 grams.
   (B) Use a suitable approximating distribution to find the probability that in at most one of these packs all 3 apples weigh at least 220 grams.
   (C) Explain why this approximating distribution is suitable.
4. 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 \(\sigma\).
   (A) Given that \(10 \%\) of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of \(\sigma\).
   (B) Sketch the distributions of the weights of both types of apple on a single diagram.

3 At a vineyard, the process used to fill bottles with wine is subject to variation. The contents of bottles are independently Normally distributed with mean \(\mu = 751.4 \mathrm { ml }\) and standard deviation \(\sigma = 2.5 \mathrm { ml }\).

1. Find the probability that a randomly selected bottle contains at least 750 ml .
2. A case of wine consists of 6 bottles. Find the probability that all 6 bottles in a case contain at least 750 ml .
3. Find the probability that, in a random sample of 25 cases, there are at least 2 cases in which all 6 bottles contain at least 750 ml . It is decided to increase the proportion of bottles which contain at least 750 ml to \(98 \%\).
4. This can be done by changing the value of \(\mu\), but retaining the original value of \(\sigma\). Find the required value of \(\mu\).
5. An alternative is to change the value of \(\sigma\), but retain the original value of \(\mu\). Find the required value of \(\sigma\).
6. Comment briefly on which method might be easier to implement and which might be preferable to the vineyard owners.

3 The scores, \(X\), in Paper 1 of an English examination have an underlying Normal distribution with mean 76 and standard deviation 12. The scores are reported as integer marks. So, for example, a score for which \(75.5 \leqslant X < 76.5\) is reported as 76 marks.

1. Find the probability that a candidate's reported mark is 76 .
2. Find the probability that a candidate's reported mark is at least 80 .
3. Three candidates are chosen at random. Find the probability that exactly one of these three candidates' reported marks is at least 80 . The proportion of candidates who receive an A* grade (the highest grade) must not exceed \(10 \%\) but should be as close as possible to \(10 \%\).
4. Find the lowest reported mark that should be awarded an A* grade. The scores in Paper 2 of the examination have an underlying Normal distribution with mean \(\mu\) and standard deviation 12.
5. Given that \(20 \%\) of candidates receive a reported mark of 50 or less, find the value of \(\mu\).

3 The wing lengths of native English male blackbirds, measured in mm , are Normally distributed with mean 130.5 and variance 11.84.

1. Find the probability that a randomly selected native English male blackbird has a wing length greater than 135 mm .
2. Given that \(1 \%\) of native English male blackbirds have wing length more than \(k \mathrm {~mm}\), find the value of \(k\).
3. Find the probability that a randomly selected native English male blackbird has a wing length which is 131 mm correct to the nearest millimetre. It is suspected that Scandinavian male blackbirds have, on average, longer wings than native English male blackbirds. A random sample of 20 Scandinavian male blackbirds has mean wing length 132.4 mm . You may assume that wing lengths in this population are Normally distributed with variance \(11.84 \mathrm {~mm} ^ { 2 }\).
4. Carry out an appropriate hypothesis test, at the \(5 \%\) significance level.
5. Discuss briefly one advantage and one disadvantage of using a \(10 \%\) significance level rather than a \(5 \%\) significance level in hypothesis testing in general.

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.

3 Many types of computer have cooling fans. The random variable \(X\) represents the lifetime in hours of a particular model of cooling fan. \(X\) is Normally distributed with mean 50600 and standard deviation 3400.

1. Find \(\mathrm { P } ( 50000 < X < 55000 )\).
2. The manufacturers claim that at least \(95 \%\) of these fans last longer than 45000 hours. Is this claim valid?
3. Find the value of \(h\) for which \(99.9 \%\) of these fans last \(h\) hours or more.
4. The random variable \(Y\) represents the lifetime in hours of a different model of cooling fan. \(Y\) is Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( Y < 60000 ) = 0.6\) and \(\mathrm { P } ( Y > 50000 ) = 0.9\). Find the values of \(\mu\) and \(\sigma\).
5. Sketch the distributions of lifetimes for both types of cooling fan on a single diagram.

5 The discrete random variables \(X\) and \(Y\) are independent with \(X \sim \mathrm {~B} \left( 32 , \frac { 1 } { 2 } \right)\) and \(Y \sim \operatorname { Po } ( 28 )\).

1. Find the values of \(\mathrm { E } ( Y - X )\) and \(\operatorname { Var } ( Y - X )\).
2. State, with justification, an approximate distribution for \(Y - X\).
3. Hence find \(\mathrm { P } ( | Y - X | \geqslant 3 )\).

4 The weights of a particular variety (A) of tomato are known to be Normally distributed with mean 80 grams and standard deviation 11 grams.

1. Find the probability that a randomly chosen tomato of variety A weighs less than 90 grams. The weights of another variety (B) of tomato are known to be Normally distributed with mean 70 grams. These tomatoes are packed in sixes using packaging that weighs 15 grams.
2. The probability that a randomly chosen pack of 6 tomatoes of variety B , including packaging, weighs less than 450 grams is 0.8463 . Show that the standard deviation of the weight of single tomatoes of variety B is 6 grams, to the nearest gram.
3. Tomatoes of variety A are packed in fives using packaging that weighs 25 grams. Find the probability that the total weight of a randomly chosen pack of variety A is greater than the total weight of a randomly chosen pack of variety B .
4. A new variety (C) of tomato is introduced. The weights, \(c\) grams, of a random sample of 60 of these tomatoes are measured giving the following results. $$\Sigma c = 3126.0 \quad \Sigma c ^ { 2 } = 164223.96$$ Find a \(95 \%\) confidence interval for the true mean weight of these tomatoes.