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Magazine Chapter 9: Problem 9
Let \(X_{1}, \ldots, X_{25}\) be a sample from a normal distribution having a variance of 100. Find the rejection region for a test at level \(\alpha=.10\) of \(H_{0}: \mu=0\) versus \(H_{A}: \mu=1.5 .\) What is the power of the test? Repeat for \(\alpha=.01.\)
Short Answer
Expert verified
Rejection region for \(\alpha = 0.10\) is \(Z > 1.28\) and for \(\alpha = 0.01\) is \(Z > 2.33\). Power is 0.588 for \(\alpha = 0.10\) and 0.86 for \(\alpha = 0.01\).
Step by step solution
01
Identify the test statistic
We use the sample mean \( \bar{X} \) to form our test statistic. Since the variance is known, the test statistic is \( Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}} \), where \( \mu_0 = 0 \), \( \sigma = 10 \) (since variance = 100), and \( n = 25 \).
02
Define the rejection region for \( \alpha = 0.10 \)
For a significance level \( \alpha = 0.10 \), we find the critical value \( Z_{\alpha} \), which corresponds to a 10% area in the tail for a one-sided test. This value is approximately 1.28. Our rejection region is thus for \( Z > 1.28 \).
03
Define the rejection region for \( \alpha = 0.01 \)
For a significance level \( \alpha = 0.01 \), we find the critical value \( Z_{\alpha} \), which corresponds to a 1% area in the tail. This value is approximately 2.33. The rejection region is therefore for \( Z > 2.33 \).
04
Calculate the power for \( \alpha = 0.10 \)
To find the power, we calculate the probability that we correctly reject \( H_0 \) when \( \mu = 1.5 \): \( \beta = P(Z > Z_{\alpha} | \mu = 1.5) = P\left( \frac{\bar{X} - 0}{10/\sqrt{25}} > 1.28 | \mu = 1.5\right) \). Calculate \( \frac{1.5}{2} - 1.28 \approx 0.22 \). Thus, the power is \( P(Z > 0.22) \approx 0.588 \).
05
Calculate the power for \( \alpha = 0.01 \)
Similarly, calculate the power at \( \alpha = 0.01 \): \( \beta = P(Z > Z_{\alpha} | \mu = 1.5) = P\left( \frac{\bar{X} - 0}{2} > 2.33 | \mu = 1.5\right) \). Compute \( \frac{1.5}{2} - 2.33 \approx -1.08 \). Thus, the power is \( P(Z > -1.08) \approx 0.86 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Normal Distribution
When working with data, especially in hypothesis testing, the normal distribution plays a crucial role. It's a bell-shaped curve representing how data values spread across a mean, allowing us to understand probabilities related to random variables. In the case of this exercise, the data is assumed to follow a normal distribution, which allows us to use specific statistical methods.
- Characteristics: The normal distribution is symmetric around its mean, with data equally likely to fall on either side of the mean.
- Parameters: It's characterized by its mean (\( \mu \)) and variance (\( \sigma^2 \)). In this exercise, the variance is known to be 100.
- Why it matters: When the sample size is large enough (usually \( n \ge 30 \) is a good rule of thumb), the sample means follow a normal distribution, even if the population distribution is not normal, thanks to the central limit theorem.
Significance Level
The significance level, denoted by \( \alpha \), is a critical concept in hypothesis testing. It represents the threshold for determining when to reject the null hypothesis. In simple terms, it is the probability of making a Type I error, where you reject a true null hypothesis.
- Common Values: Typical significance levels are 0.10, 0.05, and 0.01. In this exercise, two significance levels are examined: \( \alpha = 0.10 \) and \( \alpha = 0.01 \).
- Rejection Region: The choice of \( \alpha \) affects the critical value, determining the rejection region of the test. For example, a lower \( \alpha \) means a narrower rejection region.
- Decision Making: If the test statistic falls into the rejection region, we reject the null hypothesis. Otherwise, we do not reject it.
Test Statistic
The test statistic is a standardized value calculated from sample data, used to decide whether to reject the null hypothesis. It transforms the sample data into a single statistic, the value of which can then be compared to a critical value to make a decision.
- Formula Used: The test statistic for this exercise is defined as \( Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}} \).
- Components:
- \( \bar{X} \) is the sample mean,
- \( \mu_0 \) is the hypothesized population mean under \( H_0 \),
- \( \sigma \) is the standard deviation (10 in this exercise),
- and \( n \) is the sample size (25 in this case).
- Comparison: The value of \( Z \) is compared to critical values from the Z-table to determine acceptance or rejection of the null hypothesis.
Power of the Test
The power of the test measures a hypothesis test's ability to correctly reject a false null hypothesis. It indicates the likelihood of finding a significant result if a true effect exists.
- Definition: Mathematically, the power is \( 1 - \beta \), where \( \beta \) is the probability of making a Type II error, failing to reject a false null hypothesis.
- Importance: A test with high power has a better chance of detecting an effect that exists. The higher the power, the less likely you are to dismiss an alternative hypothesis that is true.
- Calculation: In this exercise, power is found by evaluating the probability that the test statistic exceeds the critical value under the alternative hypothesis.
- Comparison: With \( \alpha = 0.10 \), the power is approximately 0.588, and with \( \alpha = 0.01 \), it is roughly 0.86. This difference shows how a smaller \( \alpha \) can actually increase the test's power, highlighting the trade-off between \( \alpha \) size and power.
