Question

List the assumptions necessary for each of the following inferential techniques:

a. Large-sample inferences about the difference\({\mu _1} - {\mu _2}\) between population means using a two sample z-statistic

Short answer

a)

• The two samples are independent.
• The populations have any distribution.
• The two samples are large sample. i.e. \({n_1} \ge 30\,and\,{n_2} \ge 30\)

Step-by-step solution

Given Information

Let \({x_1}\,and\,{x_2}\) are two populations.

\({\mu _1}\)is the mean for population 1.

\({\mu _2}\)is the mean for population 2.

\({\mu _1} - {\mu _2}\) is the difference between them.

Test statistic and confidence interval of large sample

The test statistic is given by

\(z = \frac{{\left( {{{\bar x}_1} - {{\bar x}_2}} \right) - {D_0}}}{{\sqrt {\frac{{\sigma _1^2}}{{{n_1}}} + \frac{{\sigma _2^2}}{{{n_2}}}} }}\)

And the confidence interval is given by

\(\left( {{{\bar x}_1} - {{\bar x}_2}} \right) \pm {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\sigma _1^2}}{{{n_1}}} + \frac{{\sigma _2^2}}{{{n_2}}}} \)

Assumptions

• The two samples are independent.
• The populations have any distribution.
• The two samples are large sample. i.e. \({n_1} \ge 30\,and\,{n_2} \ge 30\)