FAD1015: Mathematics III — Tutorial 7

Centre for Foundation Studies in Science
Universiti Malaya
Session 2024/2025

Topic: Normal Distribution

Question 1

If $Z \sim N(0, 1)$, find the following probabilities.

(a) $P(Z > -0.45)$

(b) $P(Z < 2.10)$

(c) $P(Z > -3.28)$

(d) $P(0 < Z < 2.10)$

(e) $P(-2.25 < Z < -0.75)$

(f) $P(Z > 1.55)$

(g) $P(-0.45 < Z < 2.10)$

(h) $P(Z > 1.86)$

(i) $P(Z < 2)$

Question 2

If the random variable X follows a normal distribution with mean = 100 and standard deviation = 5, find the probability that:

(a) $P(X > 107)$

(b) $P(X > 98)$

(c) $P(X < 95)$

(d) $P(101 < X < 109)$

(e) $P(91 < X < 94)$

(f) $P(93 < X < 106)$

Question 3

Given $X \sim N(50, 10^2)$

(a) What are the values of the mean and standard deviation?

(b) What value of x has a z-score of 1.4?

(c) What is the z-score that corresponds to x = 30?

(d) What is the difference between positive and negative z values?

Question 4

The average playing time of CDs in a large collection is 35 minutes, and the standard deviation is 5 minutes. Use the empirical rule to find the value (in minutes):

(a) 1 standard deviation above the mean

(b) 1 standard deviation below the mean

(c) 2 standard deviations from the mean

Assuming the distribution of time is approximately normal, about what percentage of times are:

(d) between 25 and 45 minutes?

(e) less than 20 minutes or greater than 50 minutes?

(f) less than 20 minutes?

Question 5

Suppose your statistics lecturer returned your first midterm exam with only a z-score written on it. She also told you that the histogram of the scores was approximately normal. How would you interpret each of the following z-scores?

(a) 2.2

(b) -0.4

(c) 0

Question 6

The test scores of a mathematics class (FAD1015) with 800 students are distributed normally with a mean of 75 and a standard deviation of 7.

(a) What is the probability that a student chosen at random has a test score between 61 and 89?

(b) What percentage of the class has a test score between 68 and 82?

(c) Approximately how many students have a test score of more than 96?

Use the empirical rule and Z table to find the answers in 5(a)-5(c).

Question 7

In a recent study on world happiness, participants were asked to evaluate their current lives on a scale from 0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally distributed, with a mean of 5.4 and a standard deviation of 2.2. Find the probability that a randomly selected study participant's response was:

(a) less than 4

(b) between 4 and 6

(c) more than 8

Question 8

A 100-watt light bulb has an average brightness of 1640 lumens, with a standard deviation of 62 lumens.

(a) What is the probability that a 100-watt light bulb will have a brightness more than 1800 lumens?

(b) What is the probability that a 100-watt light bulb will have a brightness less than 1550 lumens?

(c) What is the probability that a 100-watt light bulb will have a brightness between 1600 and 1700 lumens?

Question 9

The Alliance Bank issues Visa and Mastercard credit cards. It is estimated that the balance on all Visa credit cards issued by the bank have a mean of RM 845 and a standard deviation of RM 270. Assume that the balances on all these Visa cards follow a normal distribution.

(a) What is the probability that a randomly selected Visa card issued by this bank has a balance between RM 1000 and RM 1400?

(b) What percentage of the Visa cards issued by this bank has a balance of RM 750 or more?

Question 10

The times, in minutes, taken by a group of students to complete a mock Biology Exam are thought to be Normally distributed with a standard deviation of 12. Find the mean time if 2.5% of the students took over 94 minutes to complete this mock Biology Exam.

Question 11

In a doctor's surgery 1 in 40 patients wait more than 45 minutes to be seen, while 9 in 10 patients wait more than 20 minutes.

(a) Assuming that these waiting times can be modelled by a Normal distribution, find the mean and standard deviation of the distribution, giving both answers correct to the nearest minute.

(b) Using the answers of part (a) determine the probability that a randomly chosen patient waits less than 18 minutes.

Question 12

Find the probability of obtaining between 4 and 15 heads, inclusive, with 32 tosses of a fair coin:

(a) using the binomial distribution formula

(b) using the normal approximation to the binomial distribution

Question 13

A golfer hits a driver shot in the fairway 72% of the time. What is the probability that:

(a) he hits 43 fairway drives in 54 total driver shots?

(b) he hits at most 39 fairway drives in 54 total driver shots?

(c) he hits more than 41 fairway drives in 54 total driver shots?

Question 14

It is known that at a certain college, 20% of the students go to college by car.

(a) If 20 students from the college are randomly selected, find the probability that:

 i) less than 9 students go to college by car

 ii) exactly 6 students go to college by car

(b) If 100 students from the college are randomly selected:

 i) what is the probability that 11 to 17 students go to college by car?

 ii) what is the value of c such that the probability of more than c students going to college by car is 0.02?

Question 15

A study found that the probability of a new herbal remedy successfully curing a type of disease is 0.75.

(a) If five patients are given the herbal remedy, find the probability that:

 i) exactly 4 patients are cured after taking the herb,

 ii) at least one patient is cured after taking the herb.

(b) If 700 patients are given the herbal remedy, find the:

 i) probability that more than 540 patients are cured after taking the herb,

 ii) integer value of m such that at least m patients are cured after taking the herb is 0.7.

Related Concepts

  • Probability Distributions — overview of statistical distributions
  • Normal Distribution — bell-shaped probability distribution
  • Standard Normal Distribution — N(0, 1)
  • Z-Score — standardized value indicating distance from mean
  • Empirical Rule — 68-95-99.7 rule
  • Binomial Distribution — discrete probability distribution
  • Normal Approximation to Binomial — using normal distribution for binomial probabilities

Related Lectures

Related Course Page

  • FAD1015 - Mathematics III

Source: FAD1015 24-25 Tutorial 7 Questions.pdf