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Optical fibers are used in telecommunications to transmit light. Suppose current technology allows production of fibers that transmit light about \(50 \mathrm{~km}\). 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 \(H_{a}: \mu>50\), with \(\mu\) denoting the mean 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 test with significance level \(.05\) is employed: \(\begin{array}{llll}\text { i. } 52 & \text { ii. } 55 & \text { iii. } 60 & \text { iv. } 70\end{array}\) b. What happens to \(\beta\) in each of the cases in Part (a) if \(\sigma\) is actually larger than \(10 ?\) Explain your reasoning.

Step by step solution

01

Understanding Type II error

A Type II error occurs when the null hypothesis is false but it is not rejected. The probability of committing a Type II error is denoted by \(\beta\). In this scenario, a Type II error would mean that the mean transmission distance of the new optical fiber is actually more than 50 km but the test fails to reject \(H_{0}\) and incorrectly concludes that the mean is 50 km.

02

Equations for calculating \(\beta\) and the test statistic

Let's denote \(Z_{\alpha}\) as the critical value from the standard normal table associated with a significance level of 0.05. We also know the standard deviation \(\sigma\) is 10 and sample size \(n\) is 10. We will use the following equations to calculate \(\beta\): 1. Test statistic: \(Z = \frac{\mu - \mu_{0}}{\frac{\sigma}{\sqrt{n}}}\) 2. \(\beta = P(Z < Z_{\alpha} - Z)\) where \(Z\) is the test statistic calculated above.

03

Calculate \(\beta\) for each case in part (a)

We use the given alternative values of \(\mu\) and substitute them into the formulas to calculate \(\beta\). For \(\mu = 52\): We use the test statistic formula: \(Z = \frac{52 - 50}{\frac{10}{\sqrt{10}}}\) Use this \(Z\) value and \(Z_{\alpha} = 1.645\) (for significance level = 0.05) in the \(\beta\) formula to get: \(\beta = P(Z < 1.645 - Z)\) Repeat the process for each value of \(\mu\)

04

Understanding the effect of increased \(\sigma\) on \(\beta\)

Looking at our formulation of the test statistic, we see that \(\sigma\) is in the denominator. This means that an increased standard deviation would reduce the test statistic \(Z\), pushing it closer to the null mean value of 50. As a result, the probability of a Type II error \(\beta\) will increase, because it becomes harder to reject the null hypothesis when the sample mean is less different from the null mean. Therefore, larger \(\sigma\) will lead to larger \(\beta\).

05

Conclusion

Therefore, \(\beta\) represents the probability of incorrectly concluding that there is no significant improvement in the new optical fiber when there actually is. This probability decreases as the actual mean transmission distance increases. Conversely, the \(\beta\) values increase if the variability in the transmission distance increases (\(\sigma\) increases), making it harder to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type II Error

In hypothesis testing, a Type II error occurs when the test fails to reject a false null hypothesis. Imagine testing a new type of optical fiber that actually transmits light farther than 50 km. If we conclude otherwise, stating that the fiber transmits at 50 km, we have made a Type II error. These errors are denoted by the probability \( \beta \). The smaller the \( \beta \), the lower the chance of making a Type II error, which means our test is more powerful in detecting when the new mean is truly greater than 50 km.

Significance Level

The significance level, often denoted as \( \alpha \), is the probability of committing a Type I error. This occurs when the null hypothesis is true but is incorrectly rejected. For the optical fiber problem, we set \( \alpha \) to 0.05, meaning there's a 5% risk of saying the fiber transmits farther than 50 km when it actually doesn't. Choosing a lower \( \alpha \) reduces the chance of a Type I error, but can increase \( \beta \), leading possibly to more Type II errors.

Standard Deviation

Standard deviation \( (\sigma) \) measures the amount of variability in a set of data points. In our scenario, it indicates how much the transmission distances of the fibers differ from each other. A \( \sigma \) of 10 km means most transmission distances cluster around the mean of 50 km. A larger \( \sigma \) suggests more spread and increases \( \beta \). This is because the variability can mask differences between the sample mean and null hypothesis mean, making it tough to detect real improvements in transmission length.

Sample Size

Sample size \( (n) \) is the number of observations in a sample. It plays a crucial role in hypothesis testing, affecting both \( \alpha \) and \( \beta \). Larger sample sizes can help decrease \( \beta \), improving test sensitivity to detect real differences. In our exercise, \( n = 10 \) is relatively small, which could mean less ability to detect a true increase in the mean fiber transmission distance. Increasing \( n \) would make our test more reliable, reducing both \( \beta \) and the confidence interval for the estimated mean.