a) YES The number of df increases when sample size increases

\(\displaystyle{n}_{{1}}:\ {11},\ {n}_{{2}}:\ {12}\)

df: \(\displaystyle{n}_{{1}}\ +\ {n}_{{2}}\ -\ {2}\)

\(\displaystyle=\ {11}\ +\ {12}\ -\ {2}\)

\(\displaystyle=\ {21}\)

\(\displaystyle{n}_{{1}}\ =\ {13},\ {n}_{{2}}\ =\ {14}\)

df \(\displaystyle=\ {13}\ +\ {14}\ -{2}\)

\(\displaystyle=\ {27}\ -\ {2}\)

\(\displaystyle=\ {25}\)

b) NO Matched pains consint of two samples that are dependent.

\(\displaystyle{H}_{{0}}:\ \mu_{{d}}\ =\ {0}\)

There are the \(\mu_d:\ \mu_1-\mu_2\begin{cases}H_0: & \mu_d = 0\\A_0: & \mu_d \neq 0\end{cases}\)

c) NO. t-distribution is used for making infernces concening two population variance.(unequal variances)

d) YES for population proportion we are using

\(\displaystyle{z}\ -\ {d}{i}{s}{t}{r}{i}{b}{u}{t}{i}{o}{n}\)

\(\displaystyle{z}\ =\ {\frac{{\hat{{p}}\ -\ {p}}}{{\sqrt{{{P}{\left({1}\ -\ {p}\right)}}}{\left\lbrace{n}\right\rbrace}}}}\)

e) NO F distribution is not symmetric but it has mean of 0

f) YES when using independent samples to that the difference between two population means, pooled variance is used.

g) NO it is necesary to have equal sample sizes for the pained to test.