===== Lab 03 : Statistical significance tests with MATLAB =====

Topics:
  * t-test
   

==== HW03 Assignment ====

For the homework HW03 tasks, follow the emphasized links to get to the detailed description.

  * **[5.0 pts]** - [[#(A) t-test]], Solve the tasks specified in the detailed description.
  * **[2.0 pts]** - [[#(B) bonus]], Solve the tasks specified in the detailed description.)

Upload your report in PDF format and code zipped in a folder to [[https://cw.felk.cvut.cz/brute/student/course/962|BRUTE]] to the assignment ''L03-Matlab: Statistics''. Please name the PDF file as ''hw03_<your_fel_id>.pdf''. Include the MATLAB code snippets in the report or upload them separately.

==== Statistical tests Introduction ====
===== t-test: =====

The following code shows exam grades of two students in vectors $x$ and $y$.
Test the null hypothesis that the pairwise difference between data vectors x and y has a mean equal to zero with a significance level ($\alpha$)=0.01

**Hints**
Use of paired sample t-test as shown in the following snippet, if  and y are two samples and $\alpha=0.01$
 <code matlab>
[h,p] = ttest(x,y,'Alpha',0.01)

</code>

The value of h = 1 (or p<0.01)indicates that "t-test" **reject the null hypothesis** at the default 1% significance level.\\
Use ttest2() function for unpaired samples from uniform distributions with equal or 'unequal' standard deviations.\\
Follow the link for various conditions of t-test:
[[https://www.investopedia.com/terms/t/t-test.asp]] \\
To generate $n$ samples from normal distribution with given mean ($m$) , standard deviation ($s$)\\
<code matlab>
normrnd(m, s ,[1,n])
</code>
 
/*
===== F-test: =====

Test the null hypothesis that the data in x and y comes from distributions with the same variance.
<code matlab>
[h,p,ci,stats] = vartest2(x,y)
</code>

The returned result h = 1 indicates that vartest2 rejects the null hypothesis at the default 5% significance level.
*/
===== (A) t-test =====
Let's generate a signal with number of samples $n=30$, with mean =0.0 and standard deviation=1.0 and noises with number of samples= $m$ from normal distributions with parameters:\\
1. mean =0.5, std.dev. =1.0, $m=30$ \\
2. mean =0.5, std.dev. =2.0, $m=30$ \\
3. mean =0.5, std.dev  =2.0, $m=20$ \\
Use appropriate t-test and find out if we can detect the signal from above mentioned noises at significance level=0.05 ?\\
hint: Null hypothesis: $H0$= there is no difference between the means of signal and noise
===== (B) bonus =====
/*
A dataset contains 480 ceramic strength measurements for 2 batches of material as follows, No of observations: 240 each, means as 688.99 and                    611.1559,  std. deviation: 65 and  61 respectively. Consider default significance level=$\alpha=0.05$
We are testing null hypothesis that variance for 2 batches are equal: \\
$$H0: \sigma_1^2=\sigma_1^2$$ \\
<note tip>tip:
Find test statistic, $F=\frac{s_1^2}{s_2^2}$ ($s_1>s_2$), Use Matlab function: finv(1-$\alpha$/2,n-1,n-1) and finv($\alpha$/2,n-1,n-1) and check if based on these values to reject null hypothesis or not.
</note>
*/
We want to determine if on average class B students score 15 marks more than class A students in an exam?
We do not have information related to the variances for both classes. To perform a t-test, we randomly collect the data of 10 students from
each class. We chose significance level($\alpha$)=0.05 as the criterion for hypothesis testing.\\
Marks of 10 randomly selected students\\
Class A: 587, 602, 627, 610, 619, 622, 605, 608, 596, 592\\
Class B: 626, 643, 647, 634, 630, 649, 629, 623, 617, 607\\

hint: The null hypothesis: $\mu_{B}-\mu_{A}\leq15$\\
Find t using following formula:
$$ t=\frac{(x_B-x_A)-(\mu_B-\mu_A)}{\sqrt{(\frac{s_B^2}{n_B} + \frac{s_A^2}{n_A})}}$$\\
where, $x$ and $s$ are sample means and standard deviations. $\mu$ are population means and $n$ are total number of students picked randomly \\
To compute p-value use matlab function(one-tailed p value)\\
<code matlab>
p=1-tcdf(t,df)
</code>
with df=degrees of freedom=(total number of students randomly chosen-2)\\
Check your results also using test() given above.