Question

Optical fibers are used in telecommunications to transmit light. Current technology allows production of fibers that transmit light about \(50 \mathrm{~km}(\) Research at Rensselaer, 1984 ). Researchers are trying to develop a new type of glass fiber that will increase this distance. In evaluating a new fiber, it is of interest to test \(H_{0}: \mu=50\) versus \(H_{0}: \mu>50\), with \(\mu\) denoting the true average transmission distance for the new optical fiber. a. Assuming \(\sigma=10\) and \(n=10\), use Appendix Table 5 to find \(\beta\), the probability of a Type II error, for each of the given alternative values of \(\mu\) when a level \(.05\) test is employed: \(\begin{array}{ll}\text { i. } 52 & \text { ii. } 55\end{array}\) iii. \(60 \quad\) iv. 70 b. What happens to \(\beta\) in each of the cases in Part (a) if \(\sigma\) is actually larger than \(10 ?\) Explain your reasoning.

Short answer

The computed \(\beta\) values correspond to type II error probabilities for the given alternative hypothesis values of \(\mu\). If a larger \(\sigma\) value would occur, the probability of making a type II error would increase, evident in a higher \(\beta\) value because of a wider spread in the data.

Step-by-step solution

Compute the standard error

First, when we know both the standard deviation (\(\sigma = 10\)) and sample size (\(n = 10\)), we can compute the standard error (SE) of the mean. The SE is given by the formula: \[SE = \frac{\sigma}{\sqrt{n}}\] By substituting the given values, we get \(SE = 10/\sqrt{10} = 3.16\).

Get the critical value for a .05 level test

The critical value for a .05 level test (one-tailed as we are testing \(H_{0}: \mu>50\)) can be fetched from a Z-table, which is approximately 1.645. This value corresponds to the Z score that is the boundary for the highest 5% of values when a data set is standard normal distributed.

Compute for \( \beta \)

To compute \( \beta \), we need to standardize the alternative \( \mu \) values using the formula: \[ Z = \frac {\mu - \mu_{0}}{SE} \] Then, subtract the critical value from this Z score to find \( \beta \). Gather the values for \( \beta \) from the Z-table. Do this for each given \( \mu \) alternative.

Infer for larger \(\sigma\)

The larger the value of \(\sigma\), the larger the value of \( \beta \) will be. This is because a larger standard deviation means more spread in the data, making it harder to reject the null hypothesis. Therefore, type II errors become more likely.