91Ó°ÊÓ

Finding power by hand. Even though software is used in practice to calculate power, doing the work by hand builds your understanding. Return to the test in Example 18.6. There are \(n=10\) observations from a population with standard deviation \(\sigma=1\) and unknown mean \(\mu\). We will test $$ \begin{aligned} &H_{0}: \mu=0 \\ &H_{a}: \mu>0 \end{aligned} $$ with fixed significance level \(\alpha=0.05\). Find the power against the alternative \(\mu=0.8\) by following these steps. (a) The \(z\) test statistic is $$ \mathrm{z}=\mathrm{x}^{-}-\mu 0 \sigma / \mathrm{n}=\mathrm{x}^{-}-01 / 10=3.162 \mathrm{x}^{-}=\frac{\bar{x}-\mu_{0}}{\sigma / \sqrt{n}}=\frac{\bar{x}-0}{1 / \sqrt{10}}=3.162 \bar{x} $$ (Remember that you won't know the numerical value of \(x^{-} \bar{x}\) until you have data.) What values of \(z\) lead to rejecting \(H_{0}\) at the \(5 \%\) significance level? (b) Starting from your result in part (a), what values of \(x^{-} \bar{x}\) lead to rejecting \(H_{0}\) ? The area above these values is shaded under the top curve in Figure 18.1. (c) The power is the probability that you observe any of these values of \(\mathrm{x}^{-} \bar{x}\) when \(\mu=0.8\). This is the shaded area under the bottom curve in Figure 18.1. What is this probability?

Step by step solution

01

Identify Critical Value for Z

For a one-tailed test at a 5% significance level, we find the critical value from the standard normal distribution table. Since it's a right-tailed test, we look for the z-value corresponding to an area of 0.95. The critical z-value is approximately 1.645.

02

Define Rejection Region for Z

The rejection region for the null hypothesis will be when the calculated z-statistic is greater than 1.645. So, reject \( H_{0} \) if \( z > 1.645 \).

03

Express Rejection Region in Terms of Sample Mean

The test statistic is given by \( z = \frac{\bar{x} - 0}{1/\sqrt{10}} = \sqrt{10}\bar{x} \). To find the critical \( \bar{x} \) that leads to rejecting \( H_{0} \), set \( \sqrt{10}\bar{x} = 1.645 \) and solve for \( \bar{x} \). This gives \( \bar{x} > \frac{1.645}{\sqrt{10}} \approx 0.520 \).

04

Calculate Probability (Power) under Alternative Hypothesis

Under the alternative hypothesis \( \mu = 0.8 \), the sample mean is normally distributed with mean 0.8 and standard deviation \( \frac{1}{\sqrt{10}} \). The probability of \( \bar{x} > 0.520 \) is equivalent to finding the probability that the z-score, \( z = \frac{\bar{x} - 0.8}{1/\sqrt{10}} \), is greater than the z-score corresponding to \( \bar{x} = 0.520 \).

05

Calculate Z for Alternative Hypothesis

Using the z-score formula for the alternative hypothesis, compute \( z = \frac{0.520 - 0.8}{1/\sqrt{10}} \), which yields \( z \approx -0.886 \).

06

Find Power from Normal Distribution

The power is the probability that \( z > -0.886 \) in the standard normal distribution. This is equivalent to finding \( P(Z > -0.886) = 1 - P(Z < -0.886) \). From the z-table, \( P(Z < -0.886) \approx 0.188 \). Therefore, the power is \( 1 - 0.188 = 0.812 \).

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Key Concepts

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

z test

Understanding the z test is crucial for analyzing hypotheses about mean values, particularly when the sample size is small, but the population standard deviation is known. It is a type of hypothesis test that helps you decide whether there is enough evidence to reject a null hypothesis or not. In the z test, we compute a statistic called the z-score, which measures how far, in standard deviation units, a sample mean is from the population mean stated in the null hypothesis.

To perform a z test, we calculate:

• The difference between the sample mean (\( \bar{x} \) ) and the population mean (\( \mu \) ) from the null hypothesis.
• Divide this difference by the standard error of the mean, which is the population standard deviation (\( \sigma \) ) divided by the square root of the sample size (\( n \) ).

This z-score tells us how unusual our sample result is under the null hypothesis, with higher absolute values indicating more unusual results.

significance level

The significance level, often denoted as \( \alpha \), is a threshold used in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true. In simpler terms, it is the risk level we are willing to take for making a type I error, which occurs when we incorrectly declare an effect or difference exists.

A common significance level used in statistical tests is 5%, or 0.05. This means that there is a 5% chance of incorrectly rejecting the null hypothesis. Choosing a significance level is a critical step in hypothesis testing as it affects the conclusions drawn.
The choice of \( \alpha \) should reflect the balance between risk tolerance and the need for certainty in results. In highly sensitive contexts, such as clinical trials, a lower significance level may be chosen to minimize risk. In less critical studies, a higher \( \alpha \) could be acceptable.

null hypothesis

In hypothesis testing, the null hypothesis represents a statement of no effect or difference. It is typically denoted as \( H_{0} \). The primary goal of hypothesis testing is to determine if there is enough statistical evidence to reject this assumption.

In our context, the null hypothesis is \( \mu = 0 \), suggesting that the true population mean is zero. This hypothesis acts as a starting point for statistical testing and is assumed true until evidence suggests otherwise.

Rejecting \( H_{0} \) indicates that the collected data provide strong evidence against this assumption. However, if evidence is insufficient, we fail to reject the null hypothesis, suggesting that any observed effect might be due to random chance rather than a real effect.

alternative hypothesis

The alternative hypothesis, symbolized as \( H_{a} \), stands in contrast to the null hypothesis. It declares that there is an actual effect or difference from the state mentioned in \( H_{0} \).

In our exercise, \( H_{a}: \mu > 0 \) implies that the population mean is greater than zero. This hypothesis is accepted if the data provide sufficient evidence that the sample mean significantly deviates from zero in the specified direction.

Testing for an alternative hypothesis involves calculating statistical metrics like the z-score to determine whether the sample data support \( H_{a} \) over \( H_{0} \).
Ultimately, accepting the alternative hypothesis suggests that the observed sample provides strong evidence for real effects or differences, beyond random sampling variability.