NED UNIVERSITY OF ENGINEERING & TECHNOLOGY

TE (EE) BATCH 2024 FALL SEMESTER

Handout #4 P&S (MT-331)

NORMAL DISTRIBUTION
1. Given a standard normal distribution, find the area under the curve that lies
(a) To the left of z = −1.39.
(b) To the right of z = 1.96.
(c) Between z = −2.16 and z = −0.65.
(d) to the left of z = 1.43;
(e) to the right of z = −0.89;
(f) Between z = −0.48 and z = 1.74.

2. Given a standard normal distribution, find the value of k such that

(a) P (Z > k) = 0.2946.
(b) P (Z < k) = 0.0427.
(c) P (−0.93 < Z < k) = 0.7235.

3. Given the normally distributed variable X with mean 18 and standard deviation 2.5, find
(a) P(X <15).
(b) The value of k such that P(X <k) = 0.2236.
(c) The value of k such that P(X >k) = 0.1814.
(d) P (17 < X < 21).

4. The speed of a file transfer from a server on campus to a personal computer at a student’s home on a weekday evening is
normally distributed with a mean of 60 kilobits per second and a standard deviation of 4 kilobits per second.
(a) What is the probability that the file will transfer at a speed of 70 kilobits per second or more?
(b) What is the probability that the file will transfer at a speed of less than 58 kilobits per second?

5. The line width of for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micrometer
and a standard deviation of 0.05 micrometer.
(a) What is the probability that a line width is greater than 0.62 micrometer?
(b) What is the probability that a line width is between 0.47 and 0.63 micrometer?

6. An electrical firm manufactures light bulbs that have a life, before burn-out, that is normally distributed with mean equal to
800 hours and a standard deviation of 40 hours. Find the probability that a bulb burns between 778 and 834 hours.

7. A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms.
Assuming that the resistance follows a normal distribution and can be measured to any degree of accuracy, what percentage
of resistors will have a resistance exceeding 43 ohms?
(b) Find the percentage of resistances exceeding 43 ohms if resistance is measured to the nearest ohm.

8. The elongation of a steel bar under a particular load has been established to be normally distributed with a mean of 0.05 inch
and σ = 0.01 inch. Find the probability that the elongation is
(a) above 0.1 inch.
(b) below 0.04 inch.
(c) Between 0.025 and 0.065 inch.

9. In a human factor experimental project, it has been determined that the reaction time of a pilot to a visual stimulus is
normally distributed with a mean of 1/2 second and standard deviation of 2/5 second.
(a) What is the probability that a reaction from the pilot takes more than 0.3 second?
(b) What reaction time is that which is exceeded 95% of the time?

10. A research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then
Shumaila Usman (Assistant professor)
(Mathematics Department)
 NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
TE (EE) BATCH 2024 FALL SEMESTER
Handout #4 P&S (MT-331)

enriched with vitamins and proteins. Assuming that the lifetimes of such mice are normally distributed with a standard
deviation of 6.3 months, find the probability that a given mouse will live
(a) more than 32 months;
(b) less than 28 months;
(c) Between 37 and 49 months.

11. A multiple-choice quiz has 200 questions, each with 4 possible answers of which only 1 is correct. What is the probability that
sheer guesswork yields from 25 to 30 correct answers for the 80 of the 200 problems about which the student has no
knowledge?

12. A process yields 10% defective items. If 100 items are randomly selected from the process, what is the probability that the
number of defectives
(a) Exceeds 13?
(b) Is less than 8?

13. The probability that a patient recovers from a delicate heart operation is 0.9. Of the next 100 patients having this operation,
what is the probability that
(a) Between 84 and 95 inclusive survive?
(b) Fewer than 86 survive?

14. The amount of time consumed by an individual at a bank ATM is found to be normally distributed with mean
μ = 130 seconds and standard deviation σ = 45 seconds.
(a) What is the probability that a randomly selected individual will consume less than 100 seconds at the ATM?
(b) What is the probability that a randomly selected individual will spend between 2 to 3 minutes at the ATM?
(c) Within what length of time do 20 per cent of individuals complete their job at the ATM?
(d) What is the least amount of time required for individuals with top 5 per cent of required time?

15. A company produces component parts for engine. Parts specifications suggest that 95% of items meet specifications. The
parts are shipped to customers in lots of 100.
(a) What is the probability that more than 2 items in a given lot will be defective?
(b) What is the probability that more than 10 items in a lot will be defective?

EXPONENTIAL DISTRIBUTION

16. The length of time for one individual to be served at a cafeteria is a random variable having an exponential distribution with a
mean of 4 minutes. What is the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days?

17. Computer response time is an important application of the gamma and exponential distributions. Suppose that a study of a
certain computer system reveals that the response time, in seconds, has an exponential distribution with a mean of 3
seconds. What is the probability that response time exceeds 5 seconds?

18. The exponential distribution is frequently applied to the waiting times between successes in a Poisson process. If the number
of calls received per hour by a telephone answering service is a Poisson random variable with parameter λ = 6, we know that
the time, in hours, between successive calls has an exponential distribution with parameter β =1/6. What is the probability of
waiting more than 15 minutes between any two successive calls?

Shumaila Usman (Assistant professor)

(Mathematics Department)
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
TE (EE) BATCH 2024 FALL SEMESTER
Handout #4 P&S (MT-331)

Shumaila Usman (Assistant professor)

(Mathematics Department)