Textbook Question
In Exercises 7–10, (d) explain how you should interpret a decision that rejects the null hypothesis.
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.13
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
Verified step by step guidance1
Step 1: Understand the type of test and significance level. This is a left-tailed z-test with a significance level (α) of 0.02. A left-tailed test means the rejection region is located in the lower tail of the standard normal distribution.
Step 2: Recall the relationship between the significance level and the critical value. The critical value corresponds to the z-score that leaves an area of α (0.02) in the left tail of the standard normal distribution.
Step 3: Use a z-table or statistical software to find the z-score that corresponds to the cumulative probability of 0.02. This z-score is the critical value for the test. In MathML, the critical value can be expressed as: , where Zα is the z-score for α = 0.02.
Step 4: Define the rejection region. For a left-tailed test, the rejection region includes all z-scores less than the critical value. In MathML, the rejection region can be expressed as: .
Step 5: Visualize the graph. Draw a standard normal distribution curve, mark the critical value on the left tail, and shade the area to the left of the critical value to represent the rejection region. Label the critical value and the significance level (α = 0.02) on the graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Value
The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. For a left-tailed test, it is the z-score that corresponds to the specified level of significance (α). In this case, with α = 0.02, the critical value indicates the point below which we would reject the null hypothesis, helping to define the rejection region.
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Rejection Region
The rejection region is the range of values for which the null hypothesis is rejected. In a left-tailed test, this region lies to the left of the critical value on the z-distribution. For α = 0.02, the rejection region includes all z-scores less than the critical value, indicating that if the test statistic falls within this region, we conclude that the sample provides sufficient evidence to reject the null hypothesis.
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09:56Step 4: State Conclusion
Z-Test
A z-test is a statistical test used to determine if there is a significant difference between sample and population means when the population variance is known. It utilizes the standard normal distribution to calculate the z-score, which measures how many standard deviations an element is from the mean. In this context, the z-test is applied to assess whether the sample data significantly deviates from the hypothesized population mean under the specified significance level.
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Related Practice
Textbook Question
In Exercises 7–10, explain how you should interpret a decision that fails to reject the null hypothesis.
A nonprofit consumer organization says that the standard deviation of the starting prices of its top-rated vehicles for a recent year is no more than \$2900.
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 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
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
Textbook Question
n Exercises 1–6, the statement represents a claim. Write its complement and state which is H0 and which is Ha.
σ > 1.9