Standard two probabilities given

6 In a cycling event the times taken to complete a course are modelled by a normal distribution with mean 62.3 minutes and standard deviation 8.4 minutes.

1. Find the probability that a randomly chosen cyclist has a time less than 74 minutes.
2. Find the probability that 4 randomly chosen cyclists all have times between 50 and 74 minutes.
   In a different cycling event, the times can also be modelled by a normal distribution. \(23 \%\) of the cyclists have times less than 36 minutes and \(10 \%\) of the cyclists have times greater than 54 minutes.
3. Find estimates for the mean and standard deviation of this distribution.

6 The heights of the female students at Breven college are normally distributed:

• \(90 \%\) of the female students have heights less than 182.7 cm .
• \(40 \%\) of the female students have heights less than 162.5 cm .
  1. Find the mean and the standard deviation of the heights of the female students at Breven college.
     \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-10_2715_41_110_2008}
     \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-11_2723_35_101_20}

Ten female students are chosen at random from those at Breven college.

Find the probability that fewer than 8 of these 10 students have heights more than 162.5 cm .

3 The random variable \(X\) is the length of time in minutes that Jannon takes to mend a bicycle puncture. \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is given that \(\mathrm { P } ( X > 30.0 ) = 0.1480\) and \(\mathrm { P } ( X > 20.9 ) = 0.6228\). Find \(\mu\) and \(\sigma\).

2 Lengths of a certain type of white radish are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm } .4 \%\) of these radishes are longer than 12 cm and \(32 \%\) are longer than 9 cm . Find \(\mu\) and \(\sigma\).

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 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.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 54.1 ) = 0.5\) and \(\mathrm { P } ( X > 50.9 ) = 0.8665\). Find the values of \(\mu\) and \(\sigma\).

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 54.1 ) = 0.5\) and \(\mathrm { P } ( X > 50.9 ) = 0.8665\). Find the values of \(\mu\) and \(\sigma\).

1 The random variable \(T\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( T > 80 ) = 0.05\) and \(\mathrm { P } ( T > 50 ) = 0.75\). Find the values of \(\mu\) and \(\sigma\).

3 The continuous random variable \(T\) has mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( T < 140 ) = 0.01\) and \(\mathrm { P } ( T < 300 ) = 0.8\).

1. Assuming that \(T\) is normally distributed, calculate the values of \(\mu\) and \(\sigma\). In fact, \(T\) represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about \(10 \%\) of runners took longer than 400 minutes.
2. State with a reason whether the mean of \(T\) would be higher than, equal to, or lower than the value calculated in part (i).

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 \(\sigma\).

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.

5. The duration of the pregnancy of a certain breed of cow is normally distributed with mean \(\mu\) days and standard deviation \(\sigma\) days. Only \(2.5 \%\) of all pregnancies are shorter than 235 days and \(15 \%\) are longer than 286 days.

1. Show that \(\mu - 235 = 1.96 \sigma\).
2. Obtain a second equation in \(\mu\) and \(\sigma\).
3. Find the value of \(\mu\) and the value of \(\sigma\).
4. Find the values between which the middle \(68.3 \%\) of pregnancies lie.

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. 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 .

6. The time taken, in minutes, by children to complete a mathematical puzzle is assumed to be normally distributed with mean \(\mu\) and standard deviation \(\sigma\). The puzzle can be completed in less than 24 minutes by \(80 \%\) of the children. For \(5 \%\) of the children it takes more than 28 minutes to complete the puzzle.

1. Show this information on the Normal curve below.
2. Write down the percentage of children who take between 24 minutes and 28 minutes to complete the puzzle.
3. 1. Find two equations in \(\mu\) and \(\sigma\).
   2. Hence find, to 3 significant figures, the value of \(\mu\) and the value of \(\sigma\). A child is selected at random.
4. Find the probability that the child takes less than 12 minutes to complete the puzzle.
   \includegraphics[max width=\textwidth, alt={}, center]{ca8418eb-4d35-40f4-af40-77503327ae52-11_314_1255_1375_356}

3. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). It is known that $$\mathrm { P } ( X \leq 66 ) = 0.0359 \text { and } \mathrm { P } ( X \geq 81 ) = 0.1151 .$$

1. In the space below, give a clearly labelled sketch to represent these probabilities on a Normal curve.
2. 1. Show that the value of \(\sigma\) is 5 .
   2. Find the value of \(\mu\).
3. Find \(\mathrm { P } ( 69 \leq X \leq 83 )\).

6 The volume of shampoo, \(V\) millilitres, delivered by a machine into bottles may be modelled by a normal random variable with mean \(\mu\) and standard deviation \(\sigma\).

1. Given that \(\mu = 412\) and \(\sigma = 8\), determine:
   1. \(\mathrm { P } ( V < 400 )\);
   2. \(\mathrm { P } ( V > 420 )\);
   3. \(\mathrm { P } ( V = 410 )\).
2. A new quality control specification requires that the values of \(\mu\) and \(\sigma\) are changed so that $$\mathrm { P } ( V < 400 ) = 0.05 \quad \text { and } \quad \mathrm { P } ( V > 420 ) = 0.01$$
   1. Show, with the aid of a suitable sketch, or otherwise, that $$400 - \mu = - 1.6449 \sigma \quad \text { and } \quad 420 - \mu = 2.3263 \sigma$$
   2. Hence calculate values for \(\mu\) and \(\sigma\).

7 A machine, which cuts bread dough for loaves, can be adjusted to cut dough to any specified set weight. For any set weight, \(\mu\) grams, the actual weights of cut dough are known to be approximately normally distributed with a mean of \(\mu\) grams and a fixed standard deviation of \(\sigma\) grams. It is also known that the machine cuts dough to within 10 grams of any set weight.

1. Estimate, with justification, a value for \(\sigma\).
2. The machine is set to cut dough to a weight of 415 grams. As a training exercise, Sunita, the quality control manager, asked Dev, a recently employed trainee, to record the weight of each of a random sample of 15 such pieces of dough selected from the machine's output. She then asked him to calculate the mean and the standard deviation of his 15 recorded weights. Dev subsequently reported to Sunita that, for his sample, the mean was 391 grams and the standard deviation was 95.5 grams. Advise Sunita on whether or not each of Dev's values is likely to be correct. Give numerical support for your answers.
3. Maria, an experienced quality control officer, recorded the weight, \(y\) grams, of each of a random sample of 10 pieces of dough selected from the machine's output when it was set to cut dough to a weight of 820 grams. Her summarised results were as follows. $$\sum y = 8210.0 \quad \text { and } \quad \sum ( y - \bar { y } ) ^ { 2 } = 110.00$$ Explain, with numerical justifications, why both of these values are likely to be correct.

3. The continuous random variable \(T\) has mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( T < 140 ) = 0.01\) and \(\mathrm { P } ( T < 300 ) = 0.8\).

1. Assuming that \(T\) is normally distributed, calculate the values of \(\mu\) and \(\sigma\). In fact, \(T\) represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about \(10 \%\) of runners took longer than 400 minutes.
2. State with a reason whether the mean of \(T\) would be higher than, equal to, or lower than the value calculated in part (i).
   [0pt] [BLANK PAGE]

11 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\).

1. Find \(\mathrm { P } ( Y > \mu - \sigma )\).
2. Given that \(\mathrm { P } ( Y > 45 ) = 0.2\) and \(\mathrm { P } ( Y < 25 ) = 0.3\), determine the values of \(\mu\) and \(\sigma\). The random variables \(U\) and \(V\) have the distributions \(\mathrm { N } ( 10,4 )\) and \(\mathrm { N } ( 12,9 )\) respectively.
3. It is given that \(\mathrm { P } ( U < b ) = \mathrm { P } ( V > c )\), where \(b > 10\) and \(c < 12\). Determine \(b\) in terms of \(c\).

18 (b)
The weight, \(Y\) grams, of marmalade in a jar can be modelled as a normal variable with mean \(\mu\) and standard deviation \(\sigma\)
18 (b) (i)
18 (b) (i) \(\_\_\_\_\) \(\_\_\_\_\) \(346 - \mu = 1.96 \sigma\)
Fully justify your answer. \(\_\_\_\_\)
[0pt] [3 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
\end{tabular}}
\hline \end{tabular} \end{center} 18 (b) (ii) Given further that $$\mathrm { P } ( Y < 336 ) = 0.14$$ find \(\mu\) and \(\sigma\)
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{0abb6343-017d-4629-8a2d-cfb405dc2d14-28_2492_1721_217_150}

5. 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.