2.04f Find normal probabilities: Z transformation

4 It is known that 20\% of male giant pandas in a certain area weigh more than 121 kg and \(71.9 \%\) weigh more than 102 kg . Weights of male giant pandas in this area have a normal distribution. Find the mean and standard deviation of the weights of male giant pandas in this area.

1 The time taken, in minutes, by a ferry to cross a lake has a normal distribution with mean 85 and standard deviation 6.8.

1. Find the probability that, on a randomly chosen occasion, the time taken by the ferry to cross the lake is between 79 and 91 minutes.
2. Over a long period it is found that \(96 \%\) of ferry crossings take longer than a certain time \(t\) minutes. Find the value of \(t\).

7 The times taken by a garage to fit a tow bar onto a car have a normal distribution with mean \(m\) hours and standard deviation 0.35 hours. It is found that \(95 \%\) of times taken are longer than 0.9 hours.

1. Find the value of \(m\).
2. On one day 4 cars have a tow bar fitted. Find the probability that none of them takes more than 2 hours to fit. The times in hours taken by another garage to fit a tow bar onto a car have the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) where \(\mu = 3 \sigma\).
3. Find the probability that it takes more than \(0.6 \mu\) hours to fit a tow bar onto a randomly chosen car at this garage.

7

1. The lengths, in centimetres, of middle fingers of women in Raneland have a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(25 \%\) of these women have fingers longer than 8.8 cm and \(17.5 \%\) have fingers shorter than 7.7 cm .
   1. Find the values of \(\mu\) and \(\sigma\).
      The lengths, in centimetres, of middle fingers of women in Snoland have a normal distribution with mean 7.9 and standard deviation 0.44. A random sample of 5 women from Snoland is chosen.
   2. Find the probability that exactly 3 of these women have middle fingers shorter than 8.2 cm .
2. The random variable \(X\) has a normal distribution with mean equal to the standard deviation. Find the probability that a particular value of \(X\) is less than 1.5 times the mean.

3 The times taken, in minutes, for trains to travel between Alphaton and Beeton are normally distributed with mean 140 and standard deviation 12.

1. Find the probability that a randomly chosen train will take less than 132 minutes to travel between Alphaton and Beeton.
2. The probability that a randomly chosen train takes more than \(k\) minutes to travel between Alphaton and Beeton is 0.675 . Find the value of \(k\).

3 The distance in metres that a ball can be thrown by pupils at a particular school follows a normal distribution with mean 35.0 m and standard deviation 11.6 m .

1. Find the probability that a randomly chosen pupil can throw a ball between 30 and 40 m .
2. The school gives a certificate to the \(10 \%\) of pupils who throw further than a certain distance. Find the least distance that must be thrown to qualify for a certificate.

3 In a normal distribution, 69\% of the distribution is less than 28 and 90\% is less than 35. Find the mean and standard deviation of the distribution.

7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.

1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
   \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).

5 The length of Paulo's lunch break follows a normal distribution with mean \(\mu\) minutes and standard deviation 5 minutes. On one day in four, on average, his lunch break lasts for more than 52 minutes.

1. Find the value of \(\mu\).
2. Find the probability that Paulo's lunch break lasts for between 40 and 46 minutes on every one of the next four days.

7 In tests on a new type of light bulb it was found that the time they lasted followed a normal distribution with standard deviation 40.6 hours. 10\% lasted longer than 5130 hours.

1. Find the mean lifetime, giving your answer to the nearest hour.
2. Find the probability that a light bulb fails to last for 5000 hours.
3. A hospital buys 600 of these light bulbs. Using a suitable approximation, find the probability that fewer than 65 light bulbs will last longer than 5130 hours.

5

1. Give an example of a variable in real life which could be modelled by a normal distribution.
2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 21.0. Given that \(\mathrm { P } ( X > 10.0 ) = 0.7389\), find the value of \(\mu\).
3. If 300 observations are taken at random from the distribution in part (ii), estimate how many of these would be greater than 22.0.

4 \includegraphics[max width=\textwidth, alt={}, center]{1a10471c-5810-44ca-9353-c2c76e190a2b-2_542_876_1425_632} The random variable \(X\) has a normal distribution with mean 4.5. It is given that \(\mathrm { P } ( X > 5.5 ) = 0.0465\) (see diagram).

1. Find the standard deviation of \(X\).
2. Find the probability that a random observation of \(X\) lies between 3.8 and 4.8.

3

1. The daily minimum temperature in degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ) in January in Ottawa is a random variable with distribution \(\mathrm { N } ( - 15.1,62.0 )\). Find the probability that a randomly chosen day in January in Ottawa has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\).
2. In another city the daily minimum temperature in \({ } ^ { \circ } \mathrm { C }\) in January is a random variable with distribution \(\mathrm { N } ( \mu , 40.0 )\). In this city the probability that a randomly chosen day in January has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\) is 0.8888 . Find the value of \(\mu\).

3 The times for a certain car journey have a normal distribution with mean 100 minutes and standard deviation 7 minutes. Journey times are classified as follows: \begin{displayquote} 'short' (the shortest \(33 \%\) of times),
'long' (the longest \(33 \%\) of times),
'standard' (the remaining 34\% of times).

1. Find the probability that a randomly chosen car journey takes between 85 and 100 minutes.
2. Find the least and greatest times for 'standard' journeys. \end{displayquote}

1 \includegraphics[max width=\textwidth, alt={}, center]{6f677fc6-3ca2-4a0d-82a2-69a7cbb8574d-2_211_1169_267_488} Measurements of wind speed on a certain island were taken over a period of one year. A box-andwhisker plot of the data obtained is displayed above, and the values of the quartiles are as shown. It is suggested that wind speed can be modelled approximately by a normal distribution with mean \(\mu \mathrm { km } \mathrm { h } ^ { - 1 }\) and standard deviation \(\sigma \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Estimate the value of \(\mu\).
2. Estimate the value of \(\sigma\).

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.

3 The times taken by students to get up in the morning can be modelled by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.

1. For a random sample of 350 students, find the number who would be expected to take longer than 20 minutes to get up in the morning.
2. 'Very slow' students are students whose time to get up is more than 1.645 standard deviations above the mean. Find the probability that fewer than 3 students from a random sample of 8 students are 'very slow'.

5 The distance the Zotoc car can travel on 20 litres of fuel is normally distributed with mean 320 km and standard deviation 21.6 km . The distance the Ganmor car can travel on 20 litres of fuel is normally distributed with mean 350 km and standard deviation 7.5 km . Both cars are filled with 20 litres of fuel and are driven towards a place 367 km away.

1. For each car, find the probability that it runs out of fuel before it has travelled 367 km .
2. The probability that a Zotoc car can travel at least \(( 320 + d ) \mathrm { km }\) on 20 litres of fuel is 0.409 . Find the value of \(d\).

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.

4 The mean of a certain normally distributed variable is four times the standard deviation. The probability that a randomly chosen value is greater than 5 is 0.15 .

1. Find the mean and standard deviation.
2. 200 values of the variable are chosen at random. Find the probability that at least 160 of these values are less than 5 .

1 In a normal distribution with mean 9.3, the probability of a randomly chosen value being greater than 5.6 is 0.85 . Find the standard deviation.

5 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 82,126 )\).

1. A value of \(X\) is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84 .
2. Five independent observations of \(X\) are taken. Find the probability that at most one of them is greater than 87.
3. Find the value of \(k\) such that \(\mathrm { P } ( 87 < X < k ) = 0.3\).

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 .

1 It is given that \(X \sim \mathrm {~N} \left( 1.5,3.2 ^ { 2 } \right)\). Find the probability that a randomly chosen value of \(X\) is less than - 2.4 .

3 The amount of fibre in a packet of a certain brand of cereal is normally distributed with mean 160 grams. 19\% of packets of cereal contain more than 190 grams of fibre.

1. Find the standard deviation of the amount of fibre in a packet.
2. Kate buys 12 packets of cereal. Find the probability that at least 1 of the packets contains more than 190 grams of fibre.