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
a.
b.
c.
d.
Ch. 7 - Hypothesis Testing with One Sample
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
Not the one you use?Change textbookSelect a different textbook
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 7.3.4
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.
Left-tailed test, α=0.01, n=35
Verified step by step guidance1
Determine the degrees of freedom (df) for the t-test. The formula for degrees of freedom is df = n - 1, where n is the sample size. In this case, df = 35 - 1.
Identify the level of significance (α) for the test. Here, α = 0.01, which represents the probability of rejecting the null hypothesis when it is true.
Since this is a left-tailed test, locate the critical value of t corresponding to α = 0.01 and df = 34 using a t-distribution table or statistical software. The critical value will be negative because it is a left-tailed test.
Define the rejection region for the test. For a left-tailed test, the rejection region is t < critical value, where the critical value is the t-score found in the previous step.
Summarize the critical value and rejection region. Clearly state that if the test statistic falls within the rejection region, the null hypothesis will be rejected.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value
A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (alpha) and the distribution of the test statistic. For a left-tailed t-test, the critical value is the point below which the null hypothesis is rejected, indicating that the sample provides sufficient evidence against it.
Recommended video:
05:50
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 left-tailed test, this region is located to the left of the critical value. If the calculated test statistic falls within this region, it suggests that the observed data is unlikely under the null hypothesis, prompting researchers to consider alternative explanations.
Recommended video:
Guided course
09:56Step 4: State Conclusion
T-Test
A t-test is a statistical test used to compare the means of two groups or to compare a sample mean to a known value when the population standard deviation is unknown. It is particularly useful for small sample sizes (typically n < 30) and is based on the t-distribution. The type of t-test (one-sample, independent, or paired) depends on the data structure and research question.
Recommended video:
08:22Tukey Test
Related Practice
Textbook Question
In Exercises 7–12, find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance α.
Right-tailed test, n=27,α=0.05
Textbook Question
Stating the Null and Alternative Hypotheses In Exercises 25–30, write the claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim.
Tablets A tablet manufacturer claims that the mean life of the battery for a certain model of tablet is more than 8 hours.
Textbook Question
Identifying Type I and Type II Errors In Exercises 31–36, describe type I and type II errors for a hypothesis test of the indicated claim.
Repeat Customers A used textbook selling website claims that at least 60% of its new customers will return to buy their next textbook.
Textbook Question
Identifying the Nature of a Hypothesis Test In Exercises 37–42, state and in words and in symbols. Then determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Explain your reasoning. Sketch a normal sampling distribution and shade the area for the P-value.
Survey A polling organization reports that the number of responses to a survey mailed to 100,000 U.S. residents is not 100,000.
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= 1.95
alpha=0.08