91影视

\(\begin{array}{*{20}{l}}{\bar x = 65.6;}&{m = 32}\\{\bar y = 59.8;}&{n = 38.}\end{array}\)

Both toughness distribution are normal.

a)

\({\sigma _1} = 1.2\)for the first sample and \({\sigma _2} = 1.1. \)for the second sample.

When there is a null hypothesis.

\({H_0}:{\mu _1} - {\mu _2} = {\Delta _0}\)

The test statistic value is based on the assumption of two normal population distributions with known \({\sigma _1}\) and \({\sigma _2}\) values, as well as two independent random samples.

\(z = \frac{{(\bar x - \bar y) - {\Delta _0}}}{{\sqrt {\frac{{\sigma _1^2}}{m} + \frac{{\sigma _2^2}}{n}} }}\)

The P value for the acceptable alternative hypothesis is derived as the sufficient area under the standard normal curve.

The hypothesis of interest are \({H_0}:{\mu _1} - {\mu _2} = 5\) versus \({H_a}:{\mu _1} - {\mu _2} > 5\)

The value of the test statistic is

\(z = \frac{{65.6 - 59.8 - 5}}{{\sqrt {1.{2^2}/32 + 1.{1^2}/38} }} = 2.89\)

The area under the standard normal curve to the right of \(z\) can be used to get the \(P\) value. The \(P\) value can be calculated using the table in the appendix or software.

\(P = P(Z > 2.89) = 1 - P(Z \le 2.89) = 1 - 0.9981 = 0.0019.\)

Since

\(P = 0.0019 > 0.001 = \alpha \)

The null hypothesis should not be reject

At level \(0.001\).

b)

The type II error \(\beta \) for \({\mu _1} - {\mu _2} = {\Delta ^'}\) vary depending on the alternative hypothesis. \({H_a}:{\mu _1} - {\mu _2} > 5\) and \({\Delta ^'} = 1\) is the alternative hypothesis, in which case the type II error is.

\(\beta \left( {{\Delta ^'}} \right) = \Phi \left( {{z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }} \right)\)

Where

The type II error \(\beta \) when \({\mu _1} - {\mu _2} = 6\) is

\(\begin{array}{l}\beta (6) = \Phi \left( {3.09 - \frac{{6 - 5}}{{\sqrt {1.{2^2}/32 + 1.{1^2}/38} }}} \right)\\ = \Phi \left( {3.09 - \frac{1}{{0.2272}}} \right)\\ = \Phi ( - 1.311) = 0.095.\end{array}\)

The table in the appendix was used to determine values of \(\Phi \) and \({z_\alpha } = {z_{0.001}} = 3.09.\)

a)do not reject null hypothesis

b)\(\beta (6) = 0.095\)