This exam is 2 hours long.
Your numerical answers should be EITHER exact OR correct to 3 decimal places.
For each question, you must show all of your working steps in order to receive full points. Zero point will be given if you only provide an answer without any working steps.
Cheating is a serious offense. Students caught cheating are subject to a zero score as well as additional penalties.

## Question 1

[12 pts] Use MME with the following data
$$1,1,0,0,0,2,1,2,2,1,0,1$$
to estimate the parameters $\theta_{1}$ and $\theta_{2}$ for the distribution of $X$ whose moment generating function is given by
$$M_{X}(t)=\left(1-\theta_{1}-\theta_{2}\right)+\theta_{1} e^{t}+\theta_{2} e^{2 t}$$

## Quesulon 2

A cake maker produces special mint cakes. Assume that the distribution of the weights of these cakes is normal with mean $21.5$ grams and standard deviation $0.4$ gram.
a) [2 pts] Find the probability that the weight of a randomly selected mint cake is fewer than 21 grams.
b) [3 pts] Suppose that 8 mint cakes are selected randomly and weighed independently. Find the probability that exactly three of these cakes will weigh fewer than 21 grams.
c) [3 pts] Suppose that 37 mint cakes weighed independently and are packed into a bag for sale. Find the probability that a randomly selected bag will contain more than $11 \%$ of the cakes weighing fewer than 21 grams.
d) [7 pts] Suppose that 24 mint cakes are selected randomly and weighed independently. Find the probability that their total weight will be greater than 520 grams.

## Question 3

Use less than 30 words to answer each of the following questions:
a) $[3 \mathrm{pts}]$ What are data?
b) [3 pts] Why do we need to collect data?
c) [3 pts] What do we want to study when a regression model is used?
d) [3 pts] What do we want to know when an ANOVA model is used?
e) $[3 \mathrm{pts}]$ Comment $\bar{X}$ and $\bar{x}$.

## Question 4

Read the data from the file exam_data.txt.
b) [1 pt] What is the $R^{2}$ of the fitted line?
c) [ 2 pts] Construct $99 \%$ C.l.s for $\beta_{0}$ and $\beta_{1}$.
d) [2 pts] Test $H_{0}: \beta_{0}=0$ against $H_{1}: \beta_{0} \neq 0$ at $0.01$ level of significance. Please draw your conclusion with evidence.
e) [ $3 \mathrm{pts}$ ] Test $H_{0}: \beta_{1}=1$ against $H_{1}: \beta_{1}>1$ at $0.01$ level of significance. Please draw your conclusion with evidence.
f) [2 pts] Predict the number of confirmed cases for Jul 18 and find its residual.

## Question 6

Read the data from the file exam_data.txt.
a) [3 pts] Find the sample mean of the daily-confirmed cases for each week.
b) Suppose the daily-confirmed cases for each week follow normal distributions with equal variances. They are also assumed to be independent.
[3 pts] Test whether the means of the daily-confirmed cases in different weeks are all equal or not at a significant level of $0.03$
c) Suppose the assumption of normal distributions cannot be made. Test whether the medians of the daily-confirmed cases in different weeks are all equal or not at a significant level of $0.03 .$
(i) $\quad[3 \mathrm{pts}]$ Find the value of $\mathrm{Fr}$.
(ii) $\quad[3 \mathrm{pts}]$ Use the result in (i) to find the value of the test statistic $\mathrm{Hr}$.
(iii) [3 pts] Draw your conclusion with evidence.

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