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Chapter 18: Problem 51

You read that a statistical test at the \(\alpha=0.05\) level has probability \(0.49\) of making a Type II error when a specific alternative is true. What is the power of the test against this alternative?

Short Answer

Expert verified
The power of the test is 0.51.

Step by step solution

01

Understanding Hypothesis Testing and Errors

In hypothesis testing, a Type II error occurs when the test fails to reject the null hypothesis when the alternative hypothesis is true. Probability of a Type II error is denoted by \( \beta \). The power of the test is defined as the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true, which is \( 1 - \beta \).
02

Identifying Given Values

We are given that the probability of making a Type II error is \( \beta = 0.49 \).
03

Calculating the Power of the Test

The power of the test is calculated using the formula: \[ \text{Power} = 1 - \beta \]. Inserting the given value \( \beta = 0.49 \), we get \[ \text{Power} = 1 - 0.49 = 0.51 \].
04

Conclusion about the Power

The calculated power of the test is \( 0.51 \), indicating there is a 51% chance of correctly identifying the alternative hypothesis as true given the specific conditions of the test.

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Key Concepts

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

Type II error
A Type II error occurs in hypothesis testing when we fail to reject the null hypothesis, even though the alternative hypothesis is actually true. Basically, it means that the test says "no" when the real world says "yes." Understanding this concept involves some probability.
  • Null Hypothesis True: The test correctly fails to reject the null hypothesis.
  • Null Hypothesis False: The test incorrectly fails to reject the null hypothesis, leading to a Type II error.
The probability of making a Type II error is denoted by the Greek letter \( \beta \). The lower the value of \( \beta \), the fewer Type II errors we make. In other words, we get better at catching the truth when it actually exists.
Statistical Power
Statistical power is a critical concept in hypothesis testing that refers to the probability that a test will correctly reject a false null hypothesis. In simple terms, it measures how good the test is at detecting something when it is there. It is calculated as \( 1 - \beta \).
  • High Power: Indicates a strong capability to detect the true effect.
  • Low Power: Suggests a limited ability to reveal the true effect, increasing the risk of a Type II error.
In our example, if the probability of making a Type II error is \( 0.49 \), the statistical power is \( 0.51 \). This means that there is a 51% chance that the test will correctly identify the alternative hypothesis as true.
Null Hypothesis
The null hypothesis is a fundamental part of hypothesis testing. It represents a statement that there is no effect or difference, and is denoted by \( H_0 \). The null hypothesis is what the test initially assumes to be true.
  • Defines the default position that there is no relationship between two measured phenomena.
  • Hypothesis tests evaluate how well the data supports this hypothesis.
Rejecting the null hypothesis means suggesting that there is enough evidence to support the presence of an alternative outcome.
Alternative Hypothesis
The alternative hypothesis, denoted by \( H_1 \) or \( H_a \), acts as the contrasting counterpart to the null hypothesis. It suggests that there is an actual effect or difference present while the null hypothesis asserts there is none.
  • States that there is a statistically significant effect or relationship present.
  • Its validity is tested within the context of hypothesis testing to determine truth.
In hypothesis testing, much attention is given to **rejecting the null hypothesis** in order to provide evidence for the alternative hypothesis. The ultimate goal is to determine if the alternative hypothesis can be considered as reality given experimental data.

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Most popular questions from this chapter

Here are data on the percent change in the total mass (in tons) of wildlife in several West African game preserves in the years 1971 to \(1999:^{9}\) $$ \begin{array}{rrrrrrrrrr} \hline 1971 & 1972 & 1973 & 1974 & 1975 & 1976 & 1977 & 1978 & 1979 & 1980 \\ 2.9 & 3.1 & -1.2 & -1.1 & -3.3 & 3.7 & 1.9 & -0.3 & -5.9 & -7.9 \\ \hline 1981 & 1982 & 1983 & 1984 & 1985 & 1986 & 1987 & 1988 & 1989 & 1990 \\ -5.5 & -7.2 & -4.1 & -8.6 & -5.5 & -0.7 & -5.1 & -7.1 & -4.2 & 0.9 \\ \hline 1991 & 1992 & 1993 & 1994 & 1995 & 1996 & 1997 & 1998 & 1999 & \\ -6.1 & -4.1 & -4.8 & -11.3 & -9.3 & -10.7 & -1.8 & -7.4 & -22.9 & \\ \hline \end{array} $$ Software gives the \(95 \%\) confidence interval for the mean annual percent change as \(-6.66 \%\) to \(-2.55 \%\). There are several reasons we might not trust this interval. (a) Examine the distribution of the data. What feature of the distribution throws doubt on the validity of statistical inference? (b) Plot the percents against year. What trend do you see in this time series? Explain why a trend over time casts doubt on the condition that years 1971 to 1999 can be treated as an SRS from a larger population of years.

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.

The power of a test is important in practice because power (a) describes how well the test performs when the null hypothesis is actually true. (b) describes how sensitive the test is to violations of conditions such as Normal population distribution. (c) describes how well the test performs when the null hypothesis is actually not true.

A college administrator questions the first 50 students he meets on campus the day after final exams are over. He asks them whether they had positive, neutral, or negative overall feelings about the term that had just ended. Suggest some reasons it may be risky to act as if the first 50 students at this particular time are an SRS of all students at this college.

A writer in a medical journal says: "An uncontrolled experiment in 37 women found a significantly improved mean clinical symptom score after treatment. Methodologic flaws make it difficult to interpret the results of this study." The writer is skeptical about the significant improvement because (a) there is no control group, so the improvement might be due to the placebo effect or to the fact that many medical conditions improve over time. (b) the \(P\)-value given was \(P=0.048\), which is too large to be convincing. (c) the response variable might not have an exactly Normal distribution in the population.
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