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Magazine Chapter 8: Problem 4
Find the rejection region (for the standardized test statistic) for each hypothesis test. Identify the test as left-tailed, right-tailed, or two- tailed. a. \(\quad H 0: \mu=-62\) Vs. Ha: \(\mu \neq-62 @ \alpha=0.005\). b. \(\quad H_{0}: \mu=73\) VS. Ha: \(\mu>73 @ \alpha=0.001\). c. \(\quad H 0: \mu=1124\) VS. Ha: \(\mu<1124 @ \alpha=0.001\). d. \(\quad H 0: \mu=0.12\) VS. Ha: \(\mu \neq 0.12 @ \alpha=0.001\).
Short Answer
Expert verified
a: Two-tailed, \(Z < -2.807\) or \(Z > 2.807\); b: Right-tailed, \(Z > 3.090\); c: Left-tailed, \(Z < -3.090\); d: Two-tailed, \(Z < -3.291\) or \(Z > 3.291\).
Step by step solution
01
Identify the test type (a)
The null hypothesis is \(H_0: \mu = -62\) and the alternative hypothesis is \(H_a: \mu eq -62\). This is a two-tailed test because the alternative hypothesis includes \(\mu eq -62\).
02
Determine the rejection region (a)
For a two-tailed test with \(\alpha = 0.005\), split the significance level equally on both tails, which means \(\alpha = 0.0025\) on each tail. Use the standard normal distribution table to find the critical values that correspond to this area. The critical values are approximately \(Z = \pm 2.807\). So, the rejection region is \(Z < -2.807\) or \(Z > 2.807\).
03
Identify the test type (b)
The null hypothesis is \(H_0: \mu = 73\) and the alternative hypothesis is \(H_a: \mu > 73\). This is a right-tailed test since the alternative hypothesis includes \(\mu > 73\).
04
Determine the rejection region (b)
For a right-tailed test with \(\alpha = 0.001\), the rejection region corresponds to a Z-value where the area to the right is 0.001. The critical value from the standard normal distribution table for \(\alpha = 0.001\) is approximately \(Z = 3.090\). Hence, the rejection region is \(Z > 3.090\).
05
Identify the test type (c)
The null hypothesis is \(H_0: \mu = 1124\) and the alternative hypothesis is \(H_a: \mu < 1124\). This is a left-tailed test since the alternative hypothesis includes \(\mu < 1124\).
06
Determine the rejection region (c)
For a left-tailed test with \(\alpha = 0.001\), the rejection region corresponds to a Z-value where the area to the left is 0.001. The critical value from the standard normal distribution table for \(\alpha = 0.001\) is approximately \(Z = -3.090\). Thus, the rejection region is \(Z < -3.090\).
07
Identify the test type (d)
The null hypothesis is \(H_0: \mu = 0.12\) and the alternative hypothesis is \(H_a: \mu eq 0.12\). This is a two-tailed test because the alternative hypothesis includes \(\mu eq 0.12\).
08
Determine the rejection region (d)
For a two-tailed test with \(\alpha = 0.001\), split the significance level equally, giving \(\alpha = 0.0005\) on each tail. Using the standard normal distribution table, the critical values that correspond to this area are approximately \(Z = \pm 3.291\). Therefore, the rejection region is \(Z < -3.291\) or \(Z > 3.291\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rejection Region
In hypothesis testing, the rejection region is a critical concept that helps us decide when to reject the null hypothesis. This method simplifies decision-making by setting a threshold, or boundaries, beyond which the null hypothesis is considered implausible.
For each type of test in hypothesis testing, there is a corresponding rejection region:
For each type of test in hypothesis testing, there is a corresponding rejection region:
- For a **two-tailed test**, the rejection zone is split between the two tails of the probability distribution. Each tail captures extreme outcomes far from what we expect under the null hypothesis.For example, in test (a), with a significance level of \(\alpha = 0.005\), the rejection region is found at the extremes of the normal distribution, specifically \(Z < -2.807\) or \(Z > 2.807\).
- For a **right-tailed test**, the rejection area is only in the right tail of the distribution. This kind of test looks for outcomes significantly higher than expected under the null hypothesis. As in test (b), with \(\alpha = 0.001\), you reject the null hypothesis if \(Z > 3.090\).
- For a **left-tailed test**, the rejection area only lies in the left tail, indicating a focus on observing significantly lower outcomes. In test (c), with \(\alpha = 0.001\), the region is \(Z < -3.090\).
Standardized Test Statistic
The standardized test statistic is an essential part of hypothesis testing, guiding us in comparing our sample data to the null hypothesis.
Also known as the Z-score in this context, it measures the number of standard deviations a data point is from the mean population value specified in the null hypothesis. Here鈥檚 a brief look at what it does and why it鈥檚 critical:
Also known as the Z-score in this context, it measures the number of standard deviations a data point is from the mean population value specified in the null hypothesis. Here鈥檚 a brief look at what it does and why it鈥檚 critical:
- **Comparison tool**: It provides a common scale to compare sample means to population means, as calculated under the null hypothesis conditions, allowing for uniform interpretation across different scenarios.
- **Simple to use**: The formula for Z-score is: \[Z = \frac{(\bar{x} - \mu_0)}{(\sigma/\sqrt{n})}\]where \(\bar{x}\) is the sample mean, \(\mu_0\) is the population mean under the null hypothesis, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
- **Establishes relationship**: When the Z-score falls within the rejection region, this signifies that the sample provides sufficient evidence against the null hypothesis.
Significance Level
The significance level, denoted as \(\alpha\), is a probability threshold that defines how extreme your test statistic must be to reject the null hypothesis. It represents the maximum risk you are willing to take of making a Type I error鈥攊ncorrectly rejecting a true null hypothesis.
Here's why it's important:
Here's why it's important:
- **Pre-test decision**: The \(\alpha\) level is chosen before conducting the test, providing a benchmark against which to measure the results.
- **Expresses threshold of doubt**: A common choice is \(\alpha = 0.05\), but it can be more strict, such as \(\alpha = 0.001\), depending on the consequences of errors. For instance, tests (b), (c), and (d) in the exercise use a stringent \(\alpha = 0.001\) to minimize false positives.
- **Integral to hypothesis testing**: It helps in formulating the critical values, beyond which observed results lead to the rejection of the null hypothesis.
Critical Value
Critical values are the boundaries of the rejection region for hypothesis tests. They correspond to the chosen significance level and tell us where the rejection region begins鈥攅xcellent tools in statistical inference.
A deeper dive into critical values includes:
A deeper dive into critical values includes:
- **Calculated from distribution**: They often come from a standard normal distribution (Z-distribution) or t-distribution tables based on the test type and \(\alpha\).
- **Directly ties to significance level**: As seen in the exercise, different \(\alpha\) levels determine the critical values. For a two-tailed test with \(\alpha = 0.001\), like test (d), the critical values are around \(Z = \pm 3.291\).
- **Benchmark setting**: Critical values help establish benchmarks for determining if the observed statistic is inherently rare enough to fall into the rejection region.
