UAS 24-25 FAD1015 Mathematics III

UNIVERSITI MALAYA FOUNDATION STUDIES IN LIFE / PHYSICAL SCIENCES PROGRAMME CENTRE FOR FOUNDATION STUDIES IN SCIENCE

ACADEMIC SESSION 2024/2025 : SEMESTER 2

FAD 1015 : Mathematics III

May 2025 Time: 2 hours

INSTRUCTIONS TO CANDIDATES:

Candidate is required to answer all questions.

(This question paper consists of 6 questions on 8 printed pages)

PART A

Question 1

(a) Determine whether the following statements are TRUE or FALSE.

(i) The trial in binomial distribution is dependent.
(ii) In a Poisson distribution, events must occur independently of each other.
(iii) The Poisson distribution is a continuous probability distribution.
(iv) The area under the graph of a uniform distribution is always greater than 1.
(v) The mean and standard deviation of the exponential distribution are equal.
(vi) The exponential distribution is symmetric.

(b) You want to test if the average blood pressure of a man differs from 120 mmHg. You collects a random sample of 12 men and record their blood pressure. You want to do the t-test in R as follows:

```r
library(BSDA)

bp_data <- c(122, 121, 119, 118, 125, 124, 126, 130,
             122, 121, 119, 125)

t.test(Bp_Data, mu=120, sigma=15, alternative =
"greater than")
```

List three errors in the code.

(i) lower triangular matrix,
(ii) identity matrix, and
(iii) diagonal matrix.

[12 marks]

Question 2

(a) Fill in the blanks with the correct answers.

(i) The ________ is used to estimate the population mean.
(ii) The ＿＿＿＿＿＿ of the mean is the distribution of all possible sample means if you select all ＿＿＿＿＿＿ of a given size.

(b) Greenbull company claims that its new energy drink increases the average attention span of students. The average attention span without the drink is 42 minutes. Mr. Joe, a researcher in that company conducts an experiment on a random sample of 25 students. He finds that the sample mean and standard deviation of attention span is 44.5 minutes and 5 minutes, respectively. At 5% significance level, he wants to test whether the energy drink has significant effect on attention span.

(i) What are Mr. Joe's hypotheses for this research?
(ii) What kind of test statistic that he should use? Justify your answer.

(i) probability
(ii) non-probability

[12 marks]

PART B

Question 3

The monthly fuel price (in RM per liter) in Malaysia can be modeled as a continuous random variable $X$ with a probability density function:

$$f(x) = \begin{cases} k(4x - x^2) & \text{if } 2 \leq x \leq 3 \ 0 & \text{otherwise} \end{cases}$$

(a) Show that $k = \frac{3}{10}$.

(b) Find the probability that the fuel price in a given month is between RM 2.50 and RM 2.80.

(c) The government considers fuel subsidies if the price exceeds RM 2.70. What is the probability that the government will provide subsidies in a randomly selected month?

(d) If the fuel price remains high (above RM 2.70) for three consecutive months, it may affect the cost of goods. Based on your answer in (c), discuss whether this situation is likely to happen.

[14 marks]

Question 4

(a) On average, six out of every 50 households own a car.

(i) Calculate the probability that exactly 12 households own a car in a random sample of 100 households by using the Poisson formula.
(ii) Determine the probability that at most 26 households own a car in a random sample of 200 households by using the Poisson table.

(b) The Mathematics exam marks are normally distributed with a mean and a standard deviation of 60 and 6, respectively.

(i) If the passing mark is 45, calculate the probability that a randomly selected student passed the exam.
(ii) Given that 1500 students took the exam, determine how many of them obtained an A, assuming that a grade A requires a minimum score of 76.
(iii) If 90% of the students passed the exam, determine the minimum passing mark.

[14 marks]

Question 5

A food company claims that the mean weight of a pack of food is equal to 75 g. Six packed food samples are chosen at random and their weights are recorded as below:

$$77, 69, 78, 69, 88, 74.$$

(a) Justify whether the company would be able to conclude that the mean weight of a pack of food is differs than 75 g, at 0.10 significance level by using traditional method (critical value/rejection region).

(b) A second test is conducted and a sample of 49 packed foods are selected where the sample standard deviation and mean are 19.27 g and 78.03 g, respectively.

(i) Justify your answer at $\alpha = 0.05$ using the $p$-value method. Does your answer in (a) changed?
(ii) Construct a 95% confidence interval of the mean weight of a pack of food.

[14 marks]

Question 6

(a) Consider the following system of linear equations.

$$\begin{aligned} 4p + q &= 8 \\ p + 2r &= -3 \\ 2p + 2q - 2r &= 14 \end{aligned}$$

(i) Rewrite the linear equations in matrix form $AX = B$.
(ii) Find determinant of $A$.
(iii) Solve the system of linear equations using the Cramer's rule.

(b) Given

$$A = \begin{pmatrix} 4 & 1 & 3 \\ -3 & 2 & 4 \\ 5 & 4 & 2 \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 & 5 & 0 \\ 2 & 4 & 3 \\ 3 & -1 & 7 \end{pmatrix}.$$

Write the **R code** to perform the followings:

(i) Create the matrix $A$ in R and label the column as `col1`, `col2` and `col3`.
(ii) Find the inverse of matrix $A$.
(iii) Find $A^T B^T$.

[14 marks]

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