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What Is Power? 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 208 is a "basic" reading level for fourthgraders. I Suppose scores on the TUDA reading test for fourthgraders in your district follow a Normal distribution with standard deviation \(\sigma=40\). In 2019 the mean score for fourthgraders in your district was 219 . 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 fourthgraders in your district is still above the basic level. You will therefore test $$ \begin{aligned} &H_{0}: \mu=208 \\ &H_{a}: \mu>208 \end{aligned} $$ If the true mean score is again 219, 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 \(\mu=219\) is \(0.394 .\) a. Explain in simple language what "power \(=0.394\) " means. b. Explain why the test you plan will not adequately protect you against incorrectly deciding that average reading scores in your district are not above basic level.

Step by step solution

01

Understand Power in Hypothesis Testing

In hypothesis testing, the power of a test refers to the probability that the test will correctly reject a false null hypothesis. In this context, power is the ability of the test to detect that the average score this year is above 208 when it is, in fact, 219.

02

Interpret Power Value of 0.394

A power value of 0.394 means that there is a 39.4% chance that your test will correctly reject the null hypothesis that the mean reading score is 208 when the true mean is actually 219. This essentially means there is a 39.4% probability of detecting a true effect.

03

Consider the Implications of Low Power

A low power (such as 0.394) implies a high chance of a Type II error, which occurs when the test fails to reject the null hypothesis even though the alternative hypothesis is true. In this situation, it would mean there's a 60.6% chance of not identifying that the mean is actually greater than 208 when it is 219.

04

Evaluate Test's Effectiveness

Given the power is less than 50%, the test is not robust enough to confidently detect that average scores are above basic levels. This means there is a high probability that the test will miss the actual performance improvement, leading administrators to potentially incorrect conclusions about student performance.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a crucial concept in statistics that helps us make decisions about population parameters based on sample data. The process starts by formulating two competing statements known as the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)).

• The null hypothesis (\(H_0\)) often represents the status quo or a statement of no effect. In our example, it asserts that the average reading score is 208.
• The alternative hypothesis (\(H_a\)) suggests that there is an effect or difference we want to prove, such as the average score being above 208.

Hypothesis testing involves calculating the probability of observing the sample data, assuming \(H_0\) is true. If this probability is low, we reject \(H_0\) in favor of \(H_a\).
The aim is to determine if there's enough evidence to support that the true average score is above the basic level.
This method is robust in making data-driven decisions and conclusions.

Type II Error

A Type II error occurs when a test fails to reject the null hypothesis (\(H_0\)) even though the alternative hypothesis (\(H_a\)) is true. In simpler terms, it's like missing a needle in a haystack.
You conclude there isn’t an effect when there actually is one.

• In our context, a Type II error means concluding that the reading scores are not above 208, even when they actually are above it.
• The probability of making a Type II error is often denoted by the symbol \(\beta\).

The power of a statistical test (\(1 - \beta\)) gives us insight into the likelihood of not making a Type II error.
A low power test, like the one mentioned with a power of 0.394, indicates a 60.6% chance of failing to detect the true effect.
Recognizing and addressing the potential for Type II errors is vital to improving test accuracy and reliability.

Normal Distribution

The normal distribution is a fundamental concept in statistics describing how data values are dispersed around a mean. It is often depicted as a bell-shaped curve. In this case study, the test scores follow a normal distribution:

• We know the scores are centered around a particular mean, and this distribution pattern helps us predict probabilities and make informed decisions.
• The standard deviation (\(\sigma = 40\)) specifies the spread of scores around the mean value.

With a known distribution, we can calculate probabilities for different outcomes, which is integral to hypothesis testing.
It allows for the assessment of how likely sample outcomes are under the assumption that the null hypothesis is true, thereby guiding decision-making processes.
Understanding the normal distribution aids in executing accurate and effective hypothesis tests.

Significance Level

The significance level, often represented by \(\alpha\), is chosen before conducting a hypothesis test. It defines the threshold for rejecting the null hypothesis. Typically set at 5% (0.05), it reflects the probability of making a Type I error or wrongly rejecting \(H_0\).

• In our scenario, a 5% significance level means there is a 5% risk of falsely declaring that the average reading score is greater than 208, when it is actually not.
• This level balances the trade-off between Type I and Type II errors.

Selecting an appropriate significance level is critical. Lower significance levels reduce the chance of a Type I error but may increase the chance of making a Type II error.
Hence, understanding and choosing a suitable \(\alpha\) is essential for making informed decisions in hypothesis testing.
It influences how we interpret outcomes and align our conclusions with actual data insights.