UAS 22-23 FAD1015 Mathematics III
UNIVERSITI MALAYA FOUNDATION STUDIES IN LIFE / PHYSICAL SCIENCES PROGRAMME CENTRE FOR FOUNDATION STUDIES IN SCIENCE
ACADEMIC SESSION 2022/2023 : SEMESTER 2
FAD1015 : MATHEMATICS III
JUNE 2023 TIME : 2 HOURS
[Watermark/logo with stacked books bearing text "ILMU PUNCHA KEMAJUAN", and the Universiti Malaya coat of arms: shield with two tigers above and one below a hibiscus flower]
INSTRUCTIONS TO CANDIDATES:
Answer all questions.
(This question paper consists of 6 questions on 13 printed pages)
PART A
Question 1
(a) State the difference between discrete and continuous random variables.
[2 marks]
(b) The amount of time a service technician needs to change the oil in a car is uniformly distributed between 10 and 25 minutes. Let $X$ be the time needed to change the oil on a car.
(i) Define the random variable $X$ in words.
(ii) Write the distribution of $X$ in Mathematical notation.
(iii) What are the parameters of $X$? Describe what they represent.
[1 + 1 + 2 = 4 marks]
(c) Determine whether the statements below are TRUE or FALSE.
(i) The set of all possible outcomes for an experiment is called a sample space.
(ii) If $P(A) = P(A \cap B) + P(B' \cap A)$, then $P(B'|A) = 1 - P(B|A)$.
(iii) $A$ is independent of event $B$ if $P(A|B) = P(B)$.
(iv) $(AB)^T = A^T B^T$, where $A$ and $B$ are suitable matrices for multiplication operation.
(v) If any two rows or columns for a matrix are identical, then the value of the determinant is zero.
(vi) If $A$ is an $n \times n$ matrix, then the determinant of $A$ exist.
[6 × 1 = 6 marks]
Question 2
(a) Identify the error(s), if any, from the following R code related to calculating basic descriptive statistics:
```r
> X=Nile
> Y=Nile[1:20]
> mean(X)
> median(X)
> stdev(X)
> mean(y)
> range(y)
```
[3 marks]
(b) You are given the following R code that aims to create a scatter plot for two variables 'sepal.width' and 'sepal.length' from dataset 'dahlia.csv'. You want the plot to looks like in Figure 1.
[Scatterplot diagram titled "Scatterplot". Horizontal axis labeled "Sepal Width" with tick marks at 2.0, 2.5, 3.0, 3.5, 4.0. Vertical axis labeled "Sepal Length" with tick marks at 4.5, 5.5, 6.5, 7.5. Numerous scattered data points distributed across the plotting area.]
*Figure 1*
You run the following R code:
```
> plot(sepal.width, sepal.length, col = 'steelblue',
+ main = 'Scatterplot',
+ xlab= 'sepal width',
+ ylab = Sepal Length,
+ pch=19)
```
However, some errors may be made in the code or the figure may not be produced as you want. Hence, provide the correct code to have the desired plot.
[3 marks]
(c) A 94% confidence interval for the mean amount of coffee dispensed by a vending machine is (7.2, 7.6) oz. Interpret this confidence interval.
[2 marks]
(d) A 95% confidence interval for a population mean is (28, 35). Can you reject the null hypothesis that $\mu = 36$ at a 5% significance level? Why?
[2 marks]
(e) Given $H_0: \mu = 25$, $H_1: \mu \neq 25$ and $p\text{-value} = 0.041$. Would you reject $H_0$ at the 0.01 level of significance? Why?
[2 marks]
PART B
Question 3
(a) Meowzy and Co. have 8 lorries available for rental each day. If each rental is for the whole day and the number of demands has a mean of 4 lorries per day, find the probability that
(i) Meowzy and Co. cannot meet the demand on any one day.
(ii) less than 6 lorries are rented in a period of 3 days.
[3 + 4 = 7 marks]
(b) The time intervals between successive cars passing a certain point on a busy road have an exponential distribution with a mean of 9 minutes.
(i) Find the probability that the time interval between two successive cars is less than 7 minutes.
(ii) Find $t$ such that we can be 80% sure that the time interval will be greater than $t$.
(iii) What is the standard deviation of the time interval between two successive cars?
[3 + 3 + 1 = 7 marks]
Question 4
(a) The following data represent the duration of first marriages (in years) ending in divorce in a village population:
$$\begin{array}{cccccc} 2 & 4 & 7 & 8 & 10 & 11 \end{array}$$
(i) By sampling without replacement, list all possible samples of size $n = 2$ and calculate the mean of all sample means.
(ii) Repeat the procedure in 4(a)(i) using $n = 5$.
(iii) Identify the shape of the sampling distribution of the mean from 4(a)(i) and 4(a)(ii). Justify your answer.
(iv) Without calculating the standard deviation or variance of the sample mean, which sampling distribution has a smaller variance? Justify your answer.
[2 + 2 + 2 + 2 = 8 marks]
(b) A tire manufacturer states that the tires produced have a median lifespan of 60,000 miles and a mean lifespan of 65,000 miles. Assume the standard deviation of the tires is 9,000 miles.
(i) Write the distribution and parameters of the sample mean in mathematical notation.
(ii) What assumptions were made to answer 4(b)(i)?
(iii) If you select a sample of $n = 81$, what is the probability that the sample mean of tire lifespan is at least 90,000 miles? Subsequently, comment on the performance of the tires.
[2 + 1 + 3 = 6 marks]
Question 5
(a) The average scores of FAD1015 Mathematics III are normally distributed, with a mean of 84 and a standard deviation of 3. Using the empirical rule, sketch and label the mean and three standard deviations from the mean. Answer these questions based on your sketching:
(i) What scores fall within 68% of the distribution?
(ii) If your score is 90, how many standard deviations are you away from the mean?
(iii) How many percent of the scores are between 78 to 87?
[2 + 1 + 1 + 1 = 5 marks]
(b) The weights of Yummylicious cereal in 16 oz boxes are normally distributed with a mean of 16 oz and a standard deviation of 0.12 oz.
(i) What is the probability that a randomly selected cereal box will have at least 15.95 oz?
(ii) The production manager was concerned that his boxes of cereal had a lower weight than the expected value of 16 oz. A sample of 50 boxes was obtained, and the mean weight was found to be 15.95 oz with a standard deviation of 0.12 oz. Evaluate this situation using the traditional method of hypothesis testing at $\alpha = 0.05$.
[3 + 6 = 9 marks]
Question 6
(a) Referring to the following R code:
```r
> A<-cbind(c(1,2,0),c(2,3,-1),c(1,-1,3))
> B<-cbind(c(4,-3,-1),c(2,0,1),c(1,5,5))
> rownames(A)<-c("row1","row2","row3")
> colnames(A)<-c("Col1","Col2","Col3")
> A
```
(i) What is the output of A?
(ii) Do you think it is possible to multiply two matrices A and B? If yes, write the R code, else, give a justification.
[5 + 2 = 7 marks]
(b) Amani, Aryan and Aydan bought 3 types of cakes at Kula Cakes shop.
Table 1 shows the number of cakes bought for each type.
| | Strawberry | Cheese | Chocolate |
|--------|-----------|--------|-----------|
| Amani | 3 | 2 | 5 |
| Aryan | 2 | 3 | 1 |
| Aydan | 1 | 5 | 4 |
*Table 1*
The total price of the cakes bought by Amani, Aryan and Aydan are RM267, RM145 and RM230 respectively. By denoting $x, y$ and $z$ as the price of each strawberry, cheese and chocolate cake, write the system of linear equations in the form of matrix equation based on the above information. Hence, solve the matrix equation using Cramer's Rule and determine the price of each type of cake.
[7 marks]
END
Verbatim transcription via Kimi K2.6 vision subagents. Pages 9-10 of original PDF missing due to transcription error. Q4 Malay text translated to English. Page numbering adjusted: Q5 and Q6 moved to contiguous pages.