UAS 23-24 FAD1015 Mathematics III

UNIVERSITI MALAYA EXAMINATION FOR FOUNDATION IN LIFE / PHYSICAL SCIENCES

ACADEMIC SESSION 2023/2024 : SEMESTER 2

FAD 1015 : Mathematics III

May/June 2024 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 statements below are TRUE or FALSE.

(i) If $X$ and $Y$ are independent events, then $P(X|Y) = P(Y)$.
(ii) If $Y$ is a continuous random variable where $a$ and $b$ are constants, then $P(a \le X \le b) = P(a < X < b)$.
(iii) The distance of students' homes from school is a discrete random variable.

(b) The amount of time, $X$, women shop for an anniversary gift is exponentially distributed with the mean time equal to 10 minutes.

(i) Write the distribution of $X$ in mathematical notation.
(ii) What is the parameter of $X$? Describe what it represents.

(i) Upper triangular matrix.
(ii) Symmetrical matrix.
(iii) Diagonal matrix.

[12 marks]

Note: Question 2 was on pages 3–4 of the original PDF, which are missing from transcription.

PART B

Question 3

(a) A research indicated that the success rate of a novel vaccine in preventing a specific illness is $0.75$.

(i) If five individuals randomly selected are vaccinated, find the probability that exactly four individuals are protected from the illness using the binomial distribution table.
(ii) If $700$ individuals randomly selected are vaccinated, find the integer value of $k$ such that the probability of at least $k$ individuals being protected from the illness is $0.7$.

(b) Let $X$ be the amount of time (in minutes) a clerk spends with her customer. The time is known to have an exponential distribution with the average amount of time equals to five minutes.

(i) Find the probability that she spends two to four minutes with a randomly selected customer.
(ii) In what amount of time are half of the customers finished interacting with the clerk?
(iii) What is the standard deviation of $X$?

[14 marks]

Question 4

(a) The following data represent the number of children that is normally distributed from a randomly selected family in a population of a small village:

$$\begin{array}{cccc} 2 & 4 & 7 & 8 \end{array}$$

(i) Two sample of $n = 2$ and $n = 3$ were drawn without replacement from the above distribution. Calculate the sample means and identify the type of sampling distribution for the sample mean from the sample size of $n = 2$ and $n = 3$.
(ii) Based on (a)(i), calculate the standard deviation of the sample mean and determine which size of sampling distribution of mean has less variability?

(b) Let $X$ be the length of stay in hospitals for the patients in a country, with mean $5.5$ days and variance $2.6$ days. Because many patients stay in the hospital for considerably more days, the distribution of length of stay is strongly skewed to the left. Consider random sample of size $100$ taken from the distribution of $X$.

(i) Calculate $\bar{X}$ and $\sigma_{\bar{x}}$.
(ii) What is the probability that the mean length of stay in hospitals for the patients will be at most $4$ days?
(iii) What is the probability that the mean length of stay in hospitals for the patients will be at least $6$ days?

[14 marks]

Question 5

In 2014, the average GPA of students at PASUM was 3.37. Recently, a survey was conducted among students enrolled in a Statistics class at PASUM, asking for their current GPA. This survey received responses from 147 students, revealing an average GPA of 3.56 with a standard deviation of 0.31. Assume that the sample is random and represents all PASUM students.

(a) Is there significant evidence that there has been a change in the average GPA of PASUM students over the past decade? Test at $\alpha = 0.01$ using the traditional method (critical value/rejection region). Show the hypothesis testing steps clearly.

(b) Determine the $p$-value and compare it to $\alpha = 0.01$. What can you conclude?

(c) Construct a 95% confidence interval of the population average GPA and interpret the confidence interval. Based on your confidence interval, is it justified that the average GPA of PASUM students has changed over the last decade? Why?

[14 marks]

Question 6

(a) Referring to the following R code:

```
> a <-c(1,2,3)
> b <-c(10, 20, 30)
> c <-c(100, 200, 300)
> d <-c(1000, 2000, 3000)
> A <-cbind(a, b, c, d)
> B <-rbind(a, b, c, d)
> SUM=sum(A)
> MEAN=mean(A[1:2,])
```

(i) What is the output of SUM and MEAN?
(ii) Do you think it is possible to multiply two matrices A and B? If yes, write the R code, else, give a justification.

(b) Matrix $\boldsymbol{S}$ is given by $\boldsymbol{S} = \begin{pmatrix} 1 & 2 & 3 \ 1 & 3 & 2 \ 2 & 1 & 3 \end{pmatrix}$.

(i) Find the determinant of $\boldsymbol{S}$.
(ii) Given the cofactor matrix of $\boldsymbol{S}$ is $\begin{pmatrix} x & 1 & -5 \\ -3 & -3 & y \\ -5 & 1 & 1 \end{pmatrix}$. Determine the values of $x$ and $y$.
(iii) Obtain the adjoint matrix of $\boldsymbol{S}$. Hence, find the inverse of $\boldsymbol{S}$.

[14 marks]

END

Verbatim transcription via Kimi K2.6 vision subagents. Pages 3-4 of original PDF missing (contained Q2).