Question

a gas supplier maintains a team of engineers who are available to deal with leaks reported by customers. most reported leaks can be dealt with quickly but some require a long time. the time (excluding travelling time) taken to deal with reported leaks is found to have a mean of 65 minutes and a standard deviation of 60 minutes.
assuming that the times may be modelled by a normal distribution, estimate the probability that:
(i) it will take more than 185 minutes to deal with a reported leak.
(ii) it will take between 50 minutes and 125 minutes to deal with a reported leak.
z - score=
percentage/probability=
the board of examiners have decided that 80% of all candidates sitting a level maths will obtain a pass grade. the actual exam marks are found to be normally distributed with a mean of 45 and a standard deviation of 7.
(a) what is the lowest score a student can get on the exam to be awarded a pass grade?
(b) given that 10% of students will achieve an a*, calculate the lowest mark required to get the highest grade.
give your answers to the nearest mark.
(a) z - score for 80% below=
actual score for 80%=
(b) z - score for 10% above=
actual score for 10% above=

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean and $\sigma$ is the standard deviation.

Step2: Calculate z - score for gas leak problem (i)

Given $\mu = 65$, $\sigma=60$ and $x = 185$. Then $z=\frac{185 - 65}{60}=\frac{120}{60}=2$.
We want $P(X>185)$, which is equivalent to $P(Z > 2)$. Since the total area under the normal curve is 1, and $P(Z\leq2)$ from the standard - normal table is 0.9772. So $P(Z > 2)=1 - 0.9772=0.0228$.

Step3: Calculate z - scores for gas leak problem (ii)

For $x_1 = 50$, $z_1=\frac{50 - 65}{60}=\frac{- 15}{60}=-0.25$.
For $x_2 = 125$, $z_2=\frac{125 - 65}{60}=\frac{60}{60}=1$.
We want $P(50

Step4: Find z - score for exam problem (a)

If 80% of candidates pass, we need to find the z - score $z$ such that $P(ZUsing the formula $x=\mu+z\sigma$, with $\mu = 45$ and $\sigma = 7$, we have $x=45+0.84\times7=45 + 5.88\approx51$.

Step5: Find z - score for exam problem (b)

If 10% of students achieve an A*, we need to find the z - score $z$ such that $P(Z>z)=0.1$, or $P(Z < z)=0.9$. Looking up in the standard - normal table, $z\approx1.28$.
Using the formula $x=\mu+z\sigma$, with $\mu = 45$ and $\sigma = 7$, we have $x=45+1.28\times7=45 + 8.96\approx54$.

Answer:

(i) The probability that it will take more than 185 minutes to deal with a reported leak is 0.0228.
(ii) The probability that it will take between 50 minutes and 125 minutes to deal with a reported leak is 0.44.
(a) The lowest score a student can get on the exam to be awarded a pass grade is 51.
(b) The lowest mark required to get the highest grade (A*) is 54.