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Highway safety engineers test new road signs, hoping that increased reflectivity will make them more visible to drivers. Volunteers drive through a test course with several of the new- and old-style signs and rate which kind shows up the best. a. Is this a one-tailed or a two-tailed test? Why? b. In this context, what would a Type I error be? c. In this context, what would a Type II error be? d. In this context, what is meant by the power of the test? e. If the hypothesis is tested at the \(1 \%\) level of significance instead of \(5 \%\), how will this affect the power of the test? f. The engineers hoped to base their decision on the reactions of 50 drivers, but time and budget constraints may force them to cut back to 20 . How would this affect the power of the test? Explain.

Step by step solution

01

Identify the tail of the test

In this case, the test is one-tailed. This can be elucidated from the fact that the engineers are looking to see if the new signs are better (more visible) than the old ones. They are not interested if the new signs are worse.

02

Define a Type I error

A Type I error would occur if the new sign were concluded to be more visible when it is not in reality. In other words, if a false positive conclusion is made in favour of the new signs.

03

Define a Type II error

A Type II error would occur if the new sign were concluded to not be more visible when it truly is. In other words, a false negative conclusion is made, neglecting the superiority of the new signs.

04

Clarify the power of the test

The power of the test in this context refers to the ability of the test to correctly accept the alternative hypothesis. If new signs are actually more visible, the power of the test will determine the ability to correctly detect this.

05

Impact of significance level on the power of the test

The 1\% level of significance is stricter than a 5\% level. This means the test becomes less likely to make a Type I error (finding a difference when one does not exist). However, reducing this threshold also makes it harder for the test to detect a true difference (if one exists), hence decreasing the test's power.

06

Impact of sample size on the power of the test

Decreasing the sample size (from 50 to 20 drivers) will decrease the power of the test. A larger sample size increases statistical power, which in turn improves the probability of detecting a true effect.

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Key Concepts

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

Type I Error

In hypothesis testing, a Type I error occurs when we incorrectly reject a true null hypothesis. This kind of error is often referred to as a "false positive." In the context of the highway safety engineers' experiment, a Type I error would mean concluding that the new road signs are more visible than the old ones when, in fact, they are not.
This mistake is problematic because it can lead to unnecessary changes and implementation of inferior products. Avoiding Type I errors is crucial in maintaining the reliability of statistical conclusions. The risk of making a Type I error is typically controlled by setting a significance level, which acts as a threshold for decision-making.

Type II Error

A Type II error happens when we fail to reject a false null hypothesis, commonly known as a "false negative." In the road sign test, this would mean concluding that the new signs are not more visible than the old ones when they actually are. This error is significant because it implies missing out on a potentially valuable improvement. Not detecting true positives can result in the continuation of less effective solutions and lost opportunities for better outcomes. Understanding Type II errors helps in designing more effective tests by balancing between avoiding false positives and negatives.

Power of a Test

The power of a statistical test is its ability to correctly detect a genuine effect when one exists. It's calculated as 1 minus the probability of a Type II error. In the engineers' context, test power indicates how likely they are to correctly conclude that the new road signs indeed improve visibility if that's true.

High test power is desirable as it means a greater chance of identifying actual benefits. Several factors can influence a test's power, including sample size and significance level, which must be strategically managed to optimize test outcomes and conclusions.

Significance Level

The significance level, denoted by alpha (\( \alpha \)), is a threshold used in hypothesis testing to determine the probability of committing a Type I error. Commonly used significance levels include 0.05 (5%) and 0.01 (1%). In the case of our highway test, using a 1% significance level means the engineers demand stronger evidence to claim the new signs are better. While this reduces the probability of a Type I error, it also decreases the test's power, making it harder to detect true benefits unless they have a larger effect size.

Sample Size

Sample size plays a crucial role in hypothesis testing as it affects both the precision of the estimates and the power of the test. With the engineers looking at 50 versus 20 drivers in their study, the larger sample size provides more accurate results and higher test power.

Why? Because larger samples better represent the population, reducing the probability of both Type I and Type II errors. When limited by budget or time, reducing sample size can compromise result quality, leading to greater uncertainty in detecting real effects, hence balancing sample size remains a key concern in study design.