Question

a. Show for the upper-tailed test with \({\sigma _1}\) and \({\sigma _2}\)known that as either\(m\) or\(n\) increases, \(\beta \)decreases when \({\mu _1} - {\mu _2} > {\Delta _0}\).

b. For the case of equal sample sizes \(\left( {m = n} \right)\)and fixed \(\alpha \),what happens to the necessary sample size \(n\) as \(\beta \) is decreased, where \(\beta \) is the desired type II error probability at a fixed alternative?

Short answer

the solution is

The corresponding \({z_\beta }\) increases as the intended \(\beta \) (type II error) lowers. When the sample sizes are equivalent, \({z_\beta }\) is in the numerator of the formula for necessary sample size \(n\), indicating that when \({z_\beta }\) rises, the sample size increases as well (see exercise 13).

Step-by-step solution

show the upper-tailed test

\({H_0}:{\mu _1} - {\mu _2} = {\Delta _0}\)represents the null hypothesis. The type II error \(\beta \) for \({\mu _1} - {\mu _2} = {\Delta ^'}\) vary depending on the alternative hypothesis. The alternative hypothesis is \({H_a}:{\mu _1} - {\mu _2} > 0\), which indicates that the type II error is present

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

Where

\(\sigma = \sqrt {\frac{{\sigma _1^2}}{m} + \frac{{\sigma _2^2}}{n}} \)

When \(n\) or \(m\) goes up, \(\sigma \) goes down (because we divide with bigger number). This implies that.

\(\frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }\)

Because the numerator is positive, the value will rise. As a result, the number.

\({z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }\)

Because \(\Phi \) is a cdf of standard normal distribution, the type Il error lowers.

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

Will decreases as well.

type II error probability

The corresponding \({z_\beta }\) increases as the intended \(\beta \) (type II error) lowers. When the sample sizes are equivalent, \({z_\beta }\) is in the numerator of the formula for necessary sample size \(n\), indicating that when \({z_\beta }\) rises, the sample size increases as well (see exercise 13).

conclusion

The corresponding \({z_\beta }\) increases as the intended \(\beta \) (type II error) lowers. When the sample sizes are equivalent, \({z_\beta }\) is in the numerator of the formula for necessary sample size \(n\), indicating that when \({z_\beta }\) rises, the sample size increases as well (see exercise 13).