---
# System prepended metadata

title: Final Exam
tags: [統計學]

---

# Final Exam
###### tags: `統計學`
## Question 1
![](https://i.imgur.com/pQraGI3.jpg)
### (a)
Let Event $\text{A}$ be the event that a sample point is left-handed and Event $\text{B}$ be the event that  a sample point is color blind
$P(\text{A}\cap\text{B})=P(\text{A}\cup\text{B})-\left[P(\text{A})+P(\text{B})\right]=0.6+0.7-0.8=0.5.$  $\blacksquare$
### (b)
$P(\text{A}|\text{B})=\dfrac{P(\text{A}\cap\text{B})}{P(\text{B})}=\dfrac{0.5}{0.7}=0.71.$ $\blacksquare$
### (c\)
$\because P(\text{A}\cap\text{B})=0.5\neq P(\text{A})P(\text{B})=0.6\times0.7=0.42$
$\therefore$ Events $\text{A}$ and $\text{B}$ are NOT independent events. $\blacksquare$
### (d)
$\because P(\text{A}\cap\text{B})=0.5\neq0\quad\therefore$ Events $\text{A}$ and $\text{B}$ are NOT disjoint events. $\blacksquare$
## Question 2
## Question 3
![](https://i.imgur.com/QNXo3I5.jpg)
$X=$ prothrombin time
population standard deviation $\sigma = 15$
population mean $\mu = 15$
sample size $n=64$
significance level $\alpha=0.05$
### a)
statistical power $1-\beta$ is between $0.85$ or $0.9$
![](https://upload.wikimedia.org/wikipedia/commons/1/10/Sample_Sizes_Effect_on_Power.png)
### b)

### c\)
Yes, because the calcuations in a) and b) is unber the basis of  symmetric sample distribution.

## Question 4
![](https://i.imgur.com/vRltrnx.jpg)
sample size $n=300$, sample size of female $n_\text{F}=200$, sample size of male $n_\text{M}=100$


| Genger | Books | Wine | Sport |
| ------- | --- | --- | --- |
| Female (1)  | 150 | 20 | 30 |
| Male   (2) |  60 | 20 | 20 |

Interest: whether males and females make the same choice or not
### Part (a)
* null hypothesis: 
* alternative hypothesis:
### Part (b)

### Part (c\)
### Part (d)
### Part (e)



## Question 5
![](https://i.imgur.com/RQFa8jT.jpg)
whether 2 samples are from same population?
### Part (a)
Define subscript 1 for "chemical", 2 for "control"
- sample statistics
    * $n_1=5$, $n_2=4$
    * $\overline{X}_1=2100$, $\overline{X}_2=3850$
    * $s_1=710.6$, $s_2=888.8$
- establishing hypothesis:
    - null hypothesis: $\mu_1=\mu_2$
    - alternative hypothesis: $\mu_1\neq\mu_2$
- $t$-test statistics: $t_\text{obs} = -2.89$
    ![](https://i.imgur.com/6umdcmZ.png =400x)

### Part (b)
$\sigma$ unknown, use $t$-distribution
total degree of freedom $\text{df}=4+3-2=5$
$t_\text{obs} = -2.89$ corresponds to a tail probability of $p=1-0.9829=0.0171$
![](https://i.imgur.com/fFMZiNf.png =300x)


### Part (c\)
significance level $\alpha = .05$, critical value $t_\text{cv}=2.571$
$\because |t_\text{obs}| = 2.89 > |t_\text{cv}|=2.571$ or $\because p=0.0171<\alpha=0.05\quad\therefore$ **null hypothesis rejected**, i.e. there is **significant** difference between the two samples
Conclusion: The two random samples are **NOT** samples from the same population distribution 
### Part (d)
