2.04f Find normal probabilities: Z transformation

The distances travelled to work, \(D \mathrm {~km}\), by the employees at a large company are normally distributed with \(D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)\).

1. Find the probability that a randomly selected employee has a journey to work of more than 20 km .
2. Find the upper quartile, \(Q _ { 3 }\), of \(D\).
3. Write down the lower quartile, \(Q _ { 1 }\), of \(D\). An outlier is defined as any value of \(D\) such that \(D < h\) or \(D > k\) where $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
4. Find the value of \(h\) and the value of \(k\). An employee is selected at random.
5. Find the probability that the distance travelled to work by this employee is an outlier.
   

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2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\). It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.

3. A drinks machine dispenses coffee into cups. A sign on the machine indicates that each cup contains 50 ml of coffee. The machine actually dispenses a mean amount of 55 ml per cup and \(10 \%\) of the cups contain less than the amount stated on the sign. Assuming that the amount of coffee dispensed into each cup is normally distributed find

1. the standard deviation of the amount of coffee dispensed per cup in ml ,
2. the percentage of cups that contain more than 61 ml . Following complaints, the owners of the machine make adjustments. Only \(2.5 \%\) of cups now contain less than 50 ml . The standard deviation of the amount dispensed is reduced to 3 ml . Assuming that the amount of coffee dispensed is still normally distributed,
3. find the new mean amount of coffee per cup.
   (4)

3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students.

1. On graph paper, draw a scatter diagram to represent these data.
2. Write down, with a reason, whether the correlation coefficient between \(x\) and \(y\) is positive or negative. The data in the table can be summarised as follows. $$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
3. Find \(S _ { x y }\). The equation of the regression line of \(y\) on \(x\) is \(y = - 106.331 + b x\).
4. Find, to 3 decimal places, the value of \(b\).
5. Find, to 3 significant figures, the mean \(\bar { y }\) and the standard deviation \(s\) of the weights of this sample of students.
6. Find the values of \(\bar { y } \pm 1.96 s\).
7. Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.

7. The random variable \(X\) is normally distributed with mean 79 and variance 144 . Find

1. \(\mathrm { P } ( X < 70 )\),
2. \(\mathrm { P } ( 64 < X < 96 )\). It is known that \(\mathrm { P } ( 79 - a \leq X \leq 79 + b ) = 0.6463\). This information is shown in the figure below. \includegraphics[max width=\textwidth, alt={}, center]{df898ff4-c3ef-400c-b4f7-f4df3757941d-6_581_983_818_590} Given that \(\mathrm { P } ( X \geq 79 + b ) = 2 \mathrm { P } ( X \leq 79 - a )\),
3. show that the area of the shaded region is 0.1179 .
4. Find the value of \(b\).

7. The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm . The weights of this group of athletes are modelled by a normal distribution with mean 85 kg and standard deviation 7.1 kg . Find the probability that a randomly chosen athlete

1. is taller than 188 cm ,
2. weighs less than 97 kg .
   (2)
3. Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
4. Comment on the assumption that height and weight are independent.

The measure of intelligence, IQ, of a group of students is assumed to be Normally distributed with mean 100 and standard deviation 15.

1. Find the probability that a student selected at random has an IQ less than 91. The probability that a randomly selected student has an IQ of at least \(100 + k\) is 0.2090 .
2. Find, to the nearest integer, the value of \(k\).
   

6. The weights of bags of popcorn are normally distributed with mean of 200 g and \(60 \%\) of all bags weighing between 190 g and 210 g .

1. Write down the median weight of the bags of popcorn.
2. Find the standard deviation of the weights of the bags of popcorn. A shopkeeper finds that customers will complain if their bag of popcorn weighs less than 180 g .
3. Find the probability that a customer will complain.

6. The random variable \(X\) has a normal distribution with mean 30 and standard deviation 5 .

1. Find \(\mathrm { P } ( X < 39 )\).
2. Find the value of \(d\) such that \(\mathrm { P } ( X < d ) = 0.1151\)
3. Find the value of \(e\) such that \(\mathrm { P } ( X > e ) = 0.1151\)
4. Find \(\mathrm { P } ( d < X < e )\).

The weight, \(X\) grams, of soup put in a tin by machine \(A\) is normally distributed with a mean of 160 g and a standard deviation of 5 g .
A tin is selected at random.

1. Find the probability that this tin contains more than 168 g . The weight stated on the tin is \(w\) grams.
2. Find \(w\) such that \(\mathrm { P } ( X < w ) = 0.01\) The weight, \(Y\) grams, of soup put into a carton by machine \(B\) is normally distributed with mean \(\mu\) grams and standard deviation \(\sigma\) grams.
3. Given that \(\mathrm { P } ( Y < 160 ) = 0.99\) and \(\mathrm { P } ( Y > 152 ) = 0.90\) find the value of \(\mu\) and the value of \(\sigma\).
   

A manufacturer fills jars with coffee. The weight of coffee, \(W\) grams, in a jar can be modelled by a normal distribution with mean 232 grams and standard deviation 5 grams.

1. Find \(\mathrm { P } ( W < 224 )\).
2. Find the value of \(w\) such that \(\mathrm { P } ( 232 < W < w ) = 0.20\) Two jars of coffee are selected at random.
3. Find the probability that only one of the jars contains between 232 grams and \(w\) grams of coffee.

The length of time, \(L\) hours, that a phone will work before it needs charging is normally distributed with a mean of 100 hours and a standard deviation of 15 hours.

1. Find \(\mathrm { P } ( L > 127 )\).
2. Find the value of \(d\) such that \(\mathrm { P } ( L < d ) = 0.10\) Alice is about to go on a 6 hour journey.
   Given that it is 127 hours since Alice last charged her phone,
3. find the probability that her phone will not need charging before her journey is completed.

1. A survey of 100 households gave the following results for weekly income \(\pounds y\).

(You may use \(\sum f y ^ { 2 } = 12452\) 800)
A histogram was drawn and the class \(200 \leqslant y < 240\) was represented by a rectangle of width 2 cm and height 7 cm .

1. Calculate the width and the height of the rectangle representing the class $$320 \leqslant y < 400$$
2. Use linear interpolation to estimate the median weekly income to the nearest pound.
3. Estimate the mean and the standard deviation of the weekly income for these data. One measure of skewness is \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
4. Use this measure to calculate the skewness for these data and describe its value. Katie suggests using the random variable \(X\) which has a normal distribution with mean 320 and standard deviation 150 to model the weekly income for these data.
5. Find \(\mathrm { P } ( 240 < X < 400 )\).
6. With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie's suggestion.

3. The continuous random variable \(Y\) is normally distributed with mean 100 and variance 256 .

1. Find \(\mathrm { P } ( Y < 80 )\).
2. Find \(k\) such that \(\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516\).

5. A random variable \(X\) has a normal distribution.

1. Describe two features of the distribution of \(X\). A company produces electronic components which have life spans that are normally distributed. Only \(1 \%\) of the components have a life span less than 3500 hours and \(2.5 \%\) have a life span greater than 5500 hours.
2. Determine the mean and standard deviation of the life spans of the components. The company gives warranty of 4000 hours on the components.
3. Find the proportion of components that the company can expect to replace under the warranty.

5. A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribution with mean 90 minutes. Only \(20 \%\) of members stay for more than 125 minutes.

1. Find the standard deviation of the normal distribution.
2. Find the probability that a visit lasts less than 25 minutes. The club introduce a closing time of 10:00 pm. Tara arrives at the club at 8:00 pm.
3. Explain whether or not this normal distribution is still a suitable model for the length of her visit.

1. A scientist found that the time taken, \(M\) minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.

Find

1. \(\mathrm { P } ( M > 160 )\).
2. \(\mathrm { P } ( 150 \leqslant M \leqslant 157 )\).
3. the value of \(m\), to 1 decimal place, such that \(\mathrm { P } ( M \leqslant m ) = 0.30\).
   

5. From experience a high-jumper knows that he can clear a height of at least 1.78 m once in 5 attempts. He also knows that he can clear a height of at least 1.65 m on 7 out of 10 attempts. Assuming that the heights the high-jumper can reach follow a Normal distribution,

1. draw a sketch to illustrate the above information,
2. find, to 3 decimal places, the mean and the standard deviation of the heights the high-jumper can reach,
3. calculate the probability that he can jump at least 1.74 m .

2. The box plot in Figure 1 shows a summary of the weights of the luggage, in kg, for each musician in an orchestra on an overseas tour. \begin{figure}[h] \end{figure} The airline's recommended weight limit for each musician's luggage was 45 kg . Given that none of the musicians' luggage weighed exactly 45 kg ,

1. state the proportion of the musicians whose luggage was below the recommended weight limit. A quarter of the musicians had to pay a charge for taking heavy luggage.
2. State the smallest weight for which the charge was made.
3. Explain what you understand by the + on the box plot in Figure 1, and suggest an instrument that the owner of this luggage might play.
4. Describe the skewness of this distribution. Give a reason for your answer. One musician of the orchestra suggests that the weights of luggage, in kg, can be modelled by a normal distribution with quartiles as given in Figure 1.
5. Find the standard deviation of this normal distribution.

6. The random variable \(X\) has a normal distribution with mean 20 and standard deviation 4 .

1. Find \(\mathrm { P } ( X > 25 )\).
2. Find the value of \(d\) such that \(\mathrm { P } ( 20 < X < d ) = 0.4641\)

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 .

8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
4. State which company you would recommend. Give reasons for your answer.

7. The distances travelled to work, \(D \mathrm {~km}\), by the employees at a large company are normally distributed with \(D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)\).

1. Find the probability that a randomly selected employee has a journey to work of more than 20 km .
2. Find the upper quartile, \(Q _ { 3 }\), of \(D\).
3. Write down the lower quartile, \(Q _ { 1 }\), of \(D\). An outlier is defined as any value of \(D\) such that \(D < h\) or \(D > k\) where $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
4. Find the value of \(h\) and the value of \(k\). An employee is selected at random.
5. Find the probability that the distance travelled to work by this employee is an outlier.

The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm .

1. Find the probability that a randomly chosen adult female is taller than 150 cm .
   (3) Sarah is a young girl. She visits her doctor and is told that she is at the 60th percentile for height.
2. Assuming that Sarah remains at the 60th percentile, estimate her height as an adult. The heights of an adult male population are normally distributed with standard deviation 9.0 cm . Given that \(90 \%\) of adult males are taller than the mean height of adult females,
3. find the mean height of an adult male.

The time, in minutes, taken to fly from London to Malaga has a normal distribution with mean 150 minutes and standard deviation 10 minutes.

1. Find the probability that the next flight from London to Malaga takes less than 145 minutes. The time taken to fly from London to Berlin has a normal distribution with mean 100 minutes and standard deviation \(d\) minutes. Given that \(15 \%\) of the flights from London to Berlin take longer than 115 minutes,
2. find the value of the standard deviation \(d\). The time, \(X\) minutes, taken to fly from London to another city has a normal distribution with mean \(\mu\) minutes. Given that \(\mathrm { P } ( X < \mu - 15 ) = 0.35\)
3. find \(\mathrm { P } ( X > \mu + 15 \mid X > \mu - 15 )\).