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The Trial Urban District Assessment (TUDA) measures educational progress within participating large urban districts. TUDA gives a reading test scored from 0 to 500 . A score of 210 is a "basic" reading level for fourth-graders. 6 Suppose scores on the TUDA reading test for fourth-graders in your district follow a Normal distribution with standard deviation \(\sigma=40\). In 2013, the mean score for fourth-graders in your district was 220 . You plan to give the reading test to a random sample of 25 fourth-graders in your district this year to test whether the mean score \(\mu\) for all fourth-graders in your district is still above the basic level. You will therefore test $$ \begin{aligned} &H_{0}: \mu=210 \\ &H_{a}: \mu>210 \end{aligned} $$ If the true mean score is again 220 , on average, students are performing above the basic level. You learn that the power of your test at the \(5 \%\) significance level against the alternative \(\mu=220\) is \(0.346\). (a) Explain in simple language what "power \(=0.346\) " means. (b) Explain why the test you plan will not adequately protect you against deciding that average reading scores in your district are not above basic level.

Step by step solution

01

Understanding Power in Hypothesis Testing

The **power** of a test is the probability that the test correctly rejects a false null hypothesis. Simply put, if the null hypothesis is false, the power is the chance that your test will find evidence against it. In this exercise, a power of 0.346 means that there is a 34.6% chance of correctly rejecting the null hypothesis \(H_0: \mu = 210\) if the true mean \(\mu = 220\).

02

Evaluating Power in Context

With a power of 0.346, the probability that the test detects that the mean score \(\mu\) is greater than 210 (assuming it is actually 220) is relatively low. This means there's a high chance (around 65.4%) that the test could fail to reject the null hypothesis even if the true mean is 220, indicating the sample size or other test parameters might not be strong enough.

03

Explaining Consequences of Low Power

Since the power is only 0.346, the test has a high risk of leading to a Type II error, where we do not reject the null hypothesis even when the true mean is greater than the "basic" level (\(\mu = 220\)). This implies that the test is not very reliable in identifying whether the mean score is above the basic level, under the given conditions.

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Key Concepts

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

Power of a Test

The concept of the "power" of a test is crucial in hypothesis testing. It refers to the probability that a test will correctly reject a false null hypothesis.
This means it measures a test's ability to detect a true effect when it's present. For example, if you are testing whether students' scores are above the basic level, the power is the chance that your test will confirm they are, provided they really are above that level.
When we say that power is 0.346, it indicates that there is only a 34.6% probability that the test will successfully show the scores are above the basic level when they truly are. This value informs how reliable your test will be in catching a true positive result.
A power value closer to 1 signifies a highly effective test, while a value closer to 0 indicates a weak test. In the case mentioned, a power of 0.346 is relatively low, suggesting that improvements might be needed, such as increasing the sample size to enhance the test's effectiveness.

Type II Error

In hypothesis testing, a Type II error occurs when the test fails to reject a false null hypothesis. Essentially, it's when the test does not provide enough evidence to support the alternative hypothesis even though it should. In the context of the reading scores example provided, a Type II error would happen if we conclude that the scores are not higher than the basic level when they actually are.
The risk of committing a Type II error is inversely related to the power of the test. As noted, with the power only being 0.346, there's a significant likelihood (about 65.4%) of making a Type II error in this scenario.
This high probability of error suggests the test might not be sensitive enough in detecting differences when they exist. Common causes of higher Type II error rates include small sample sizes or minimal effect sizes. To reduce the risk of a Type II error, one could increase the sample size, enhance measurement accuracy, or use a more suitable test design.

Null Hypothesis

The "null hypothesis" in hypothesis testing is a statement suggesting no effect or no difference. It is the default or beginning assumption made about a population parameter—in this case, the mean reading score \( \mu = 210 \) being the "basic" level.
When conducting a hypothesis test, the aim is to gather enough evidence to determine whether to reject this initial assumption. The null hypothesis serves as a marker for testing and is often set up against an "alternative hypothesis," which posits that there is some significant effect or difference—in our example, that the mean score is greater than 210.
Rejecting the null hypothesis would imply that you have strong enough data to conclude that the average score is indeed above the basic level. On the other hand, failing to reject the null hypothesis suggests that the data do not provide sufficient evidence to support a score higher than 210.
The rejection or non-rejection of the null hypothesis can heavily impact decision-making in education and policy. It is crucial to use comprehensive sample sizes and appropriately designed tests to form accurate conclusions.