Textbook Question
Finding Critical Values and Rejection Regions In Exercises 23–28, find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer.
Left-tailed test, α = 0.09
Ch. 7 - Hypothesis Testing with One Sample
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 7, Problem 7.4.5
In Exercises 3–8, find the critical value(s) and rejection region(s) for the type of t-test with level of significance alpha and sample size n.
Right-tailed test, α=0.05, n=23
Verified step by step guidance1
Step 1: Identify the degrees of freedom (df) for the t-test. The degrees of freedom are calculated as df = n - 1, where n is the sample size. For this problem, df = 23 - 1.
Step 2: Recognize that this is a right-tailed t-test. This means the rejection region will be in the upper tail of the t-distribution.
Step 3: Use the level of significance (α = 0.05) and the degrees of freedom (df = 22) to find the critical value from a t-distribution table or statistical software. Look up the t-value corresponding to a cumulative probability of 1 - α = 0.95 for df = 22.
Step 4: Define the rejection region. For a right-tailed test, the rejection region consists of all t-values greater than the critical value found in Step 3.
Step 5: Summarize the results. The critical value and rejection region are now determined, and these will be used to decide whether to reject the null hypothesis in the context of the t-test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value
A critical value is a point on the scale of the test statistic that separates the region where the null hypothesis is rejected from the region where it is not rejected. In hypothesis testing, critical values are determined based on the significance level (alpha) and the distribution of the test statistic. For a right-tailed t-test, the critical value corresponds to the point where the cumulative probability equals 1 - alpha.
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Critical Values: t-Distribution
Rejection Region
The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. In a right-tailed test, this region is located to the right of the critical value. If the calculated test statistic falls within this region, it indicates that the observed data is sufficiently unlikely under the null hypothesis, prompting its rejection.
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09:56Step 4: State Conclusion
T-Test
A t-test is a statistical test used to determine if there is a significant difference between the means of two groups, or between a sample mean and a known value. It is particularly useful when the sample size is small (typically n < 30) and the population standard deviation is unknown. The t-test uses the t-distribution, which accounts for the additional uncertainty introduced by estimating the population standard deviation from the sample.
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Related Practice
Textbook Question
Finding a P-Value In Exercises 13–18, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance alpha.
Left-tailed test
z=-1.68
alpha=0.05
Textbook Question
Graphical Analysis In Exercises 21 and 22, state whether each standardized test statistic z allows you to reject the null hypothesis. Explain your reasoning.
a. z = -1.301
b. z = 1.203
c. z = 1.280
d. z = 1.286
Textbook Question
In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.
Claim: σ^2>19, α=0.1. Sample statistics: s^2=28, n=17
Textbook Question
Writing Hypotheses: Medicine A medical research team is investigating the mean cost of a 30-day supply of a heart medication. A pharmaceutical company thinks that the mean cost is less than \$60. You want to support this claim. How would you write the null and alternative hypotheses?
Textbook Question
Graphical Analysis In Exercises 17–20, match the alternative hypothesis with its graph. Then state the null hypothesis and sketch its graph.
Ha: μ < 3