Textbook Question
n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.
μ ≤ 375
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.RE.33
In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.
Left-tailed test, α=0.05, n=15
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 = 15 - 1.
Identify the level of significance (α). For this problem, α = 0.05, which represents the probability of rejecting the null hypothesis when it is true.
Since this is a left-tailed test, the critical value corresponds to the t-score where the cumulative probability to the left of the t-score equals α. Use a t-distribution table or statistical software to find the critical t-value for df = 14 and α = 0.05.
Define the rejection region. For a left-tailed test, the rejection region is t < critical value. This means any t-score less than the critical value will lead to rejecting the null hypothesis.
Summarize the critical value and rejection region. Clearly state the critical t-value and the corresponding rejection region based on the calculations or table lookup.
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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 threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α) and the distribution of the test statistic. For a left-tailed t-test, the critical value is the point on the t-distribution that corresponds to the cumulative probability of α, indicating the cutoff for the rejection region.
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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 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 sample provides sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
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Guided course
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, particularly when the sample size is small and the population standard deviation is unknown. The test uses the t-distribution, which accounts for the sample size and variability. In this case, a left-tailed t-test is employed to assess whether the sample mean is significantly less than a hypothesized population mean.
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Related Practice
Textbook Question
In Exercises 7–10, state the null and alternative hypotheses and identify which represents the claim,
An energy bar maker claims that the mean number of grams of carbohydrates in one bar is less than 25.
Textbook Question
In Exercises 29–34, find the critical value(s) and rejection region(s) for the type of t-test with level of significance α and sample size n.
Two-tailed test, α=0.02, n=12
Textbook Question
n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.
p < 0.205
Textbook Question
In Exercises 55–58, test the claim about the population variance or standard deviation at the level of significance . Assume the population is normally distributed.
Claim: σ^2 > 2; α=0.10. Sample statistics: s^2 = 2.95, n=18
Textbook Question
In Exercises 7–10, (c) explain whether the hypothesis test is left-tailed, right-tailed, or two-tailed.
An energy bar maker claims that the mean number of grams of carbohydrates in one bar is less than 25.