Answer:
a) 0.023
b) 0.286
c) 10 students will be unable to complete the exam inthe allotted time.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 80 minutes
Standard Deviation, σ = 10 minutes
We are given that the distribution of time to complete an exam is a bell shaped distribution that is a normal distribution.
Formula:
![z_{score} = \displaystyle\frac{x-\mu}{\sigma}](https://tex.z-dn.net/?f=z_%7Bscore%7D%20%3D%20%5Cdisplaystyle%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D)
a) P(completing the exam in one hour or less)
P(x < 60)
![P( x < 60) = P( z < \displaystyle\frac{60 - 80}{10}) = P(z < -2)](https://tex.z-dn.net/?f=P%28%20x%20%3C%2060%29%20%3D%20P%28%20z%20%3C%20%5Cdisplaystyle%5Cfrac%7B60%20-%2080%7D%7B10%7D%29%20%3D%20P%28z%20%3C%20-2%29)
Calculation the value from standard normal z table, we have,
![P(x < 60) =0.023= 2.3\%](https://tex.z-dn.net/?f=P%28x%20%3C%2060%29%20%3D0.023%3D%202.3%5C%25)
b) P(complete the exam in more than 60 minutes but less than 75 minutes)
![P(60 \leq x \leq 75) = P(\displaystyle\frac{60 - 80}{10} \leq z \leq \displaystyle\frac{75-80}{10}) = P(-2 \leq z \leq -0.5)\\\\= P(z \leq -0.5) - P(z < -2)\\= 0.309- 0.023 = 0.286= 28.6\%](https://tex.z-dn.net/?f=P%2860%20%5Cleq%20x%20%5Cleq%2075%29%20%3D%20P%28%5Cdisplaystyle%5Cfrac%7B60%20-%2080%7D%7B10%7D%20%5Cleq%20z%20%5Cleq%20%5Cdisplaystyle%5Cfrac%7B75-80%7D%7B10%7D%29%20%3D%20P%28-2%20%5Cleq%20z%20%5Cleq%20-0.5%29%5C%5C%5C%5C%3D%20P%28z%20%5Cleq%20-0.5%29%20-%20P%28z%20%3C%20-2%29%5C%5C%3D%200.309-%200.023%20%3D%200.286%3D%2028.6%5C%25)
c) P(completing the exam in more than 90 minutes)
P(x > 90)
![P( x > 90) = P( z > \displaystyle\frac{90 -80}{10}) = P(z > 1)](https://tex.z-dn.net/?f=P%28%20x%20%3E%2090%29%20%3D%20P%28%20z%20%3E%20%5Cdisplaystyle%5Cfrac%7B90%20-80%7D%7B10%7D%29%20%3D%20P%28z%20%3E%201%29)
![= 1 - P(z \leq 1)](https://tex.z-dn.net/?f=%3D%201%20-%20P%28z%20%5Cleq%201%29)
Calculation the value from standard normal z table, we have,
![P(x > 90) = 1 - 0.8413 = 0.1587 = 15.87\%](https://tex.z-dn.net/?f=P%28x%20%3E%2090%29%20%3D%201%20-%200.8413%20%3D%200.1587%20%3D%2015.87%5C%25)
15.87% of children of class will require more than 90 minutes to complete the test.
Number of children =
![\dfrac{15.87}{100}\times 60 = 9.52\approx 10](https://tex.z-dn.net/?f=%5Cdfrac%7B15.87%7D%7B100%7D%5Ctimes%2060%20%3D%209.52%5Capprox%2010)
Approximately, 10 students of class will require more than 90 minutes to complete the test.