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.
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.29
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=0.05, n=20
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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, n = 20, so df = 20 - 1.
Identify the level of significance (α) for the test. For a two-tailed test, the significance level is split equally between the two tails of the t-distribution. Here, α = 0.05, so each tail will have α/2 = 0.025.
Use a t-distribution table or statistical software to find the critical t-value(s) corresponding to df = 19 and a cumulative probability of 1 - α/2 = 0.975 for the upper tail and α/2 = 0.025 for the lower tail.
The rejection region(s) for the two-tailed test are defined as the areas in the tails of the t-distribution where the test statistic falls beyond the critical t-values. Specifically, the rejection regions are t < -t_critical and t > t_critical.
Summarize the critical values and rejection regions. For example, if the critical t-values are ±t_critical, the rejection regions are t < -t_critical and t > t_critical. These regions indicate where the null hypothesis would be rejected.
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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 beyond which we reject the null hypothesis. In a two-tailed test, there are two critical values that correspond to the significance level (α). For example, with α = 0.05, the critical values define the boundaries of the rejection regions in both tails of the distribution.
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Critical Values: t-Distribution
Rejection Region
The rejection region is the area in the tails of the probability distribution where we reject the null hypothesis. For a two-tailed test with α = 0.05, the rejection regions are located in both tails, each containing 2.5% of the total probability. If the test statistic falls within these regions, we conclude that the observed data is statistically significant.
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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, especially when the sample size is small (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
In Exercises 11 and 12, find the P-value for the hypothesis test with the standardized test statistic z. Decide whether to reject H0 for the level of significance α.
Left-tailed test, z = -0.94, α = 0.05
Textbook Question
In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.
Left-tailed test, α=0.02
Textbook Question
In Exercises 45–48, determine whether a normal sampling distribution can be used to approximate the binomial distribution. If it can, test the claim.
Claim: p≥0.04; α=0.10
Sample statistics: p_hat = 0.03, n=30
Textbook Question
In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.
Two-tailed test, α=0.03
Textbook Question
In Exercises 13–16, find the critical value(s) and rejection region(s) for the type of z-test with level of significance . Include a graph with your answer.
Right-tailed test, α=0.025