Difference random variable of 2 variables

Description:

• The variable is made from minusing one datapoint to the otehr

When $\sigma$ is known:

• When $n$ tends to larger, ${\bar{x}}_1-{\bar{x}}_2\sim N(D_0,\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}})$
  • $D_0={\mu }_1-{\mu }_2$

When $\sigma$ is unknown:

• When $n$ tends to larger, ${\bar{x}}_1-{\bar{x}}_2\sim T(D_0,\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}})$
  • with df

Match samples:

• To see when the 2 samples have the same mean by collecting $n$ samples from each
• Let $d=x_a-x_b$
  • $\to \bar{d}=\frac{\sum d_i}{n},s_d=\sqrt{\frac{\sum (d_i-\bar{d}{)}^2}{n-1}}$
  • $\to H_0:{\mu }_d=0,H_1:{\mu }_d\ne 0$

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