Topics:

• t-test

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 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.

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 $\alpha=0.01$

[h,p] = ttest(x,y,'Alpha',0.01)

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 ($s$)

normrnd(m, s ,[1,n])

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

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)

p=1-tcdf(t,df)

with df=degrees of freedom=(total number of students randomly chosen-2)
Check your results also using test() given above.