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.12
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.
Two-tailed test, z = 2.57, α = 0.10
Verified step by step guidance1
Step 1: Understand the problem. This is a two-tailed hypothesis test where the standardized test statistic z = 2.57, and the level of significance α = 0.10. The goal is to find the P-value and decide whether to reject the null hypothesis H₀.
Step 2: Recall that for a two-tailed test, the P-value is calculated as the area in both tails of the standard normal distribution beyond the absolute value of the test statistic z. Mathematically, P-value = 2 × P(Z > |z|).
Step 3: Use the standard normal distribution table (or a statistical software) to find the probability P(Z > 2.57). This represents the area to the right of z = 2.57 under the standard normal curve.
Step 4: Multiply the result from Step 3 by 2 to account for both tails of the distribution, as this is a two-tailed test. This gives the total P-value.
Step 5: Compare the P-value obtained in Step 4 with the level of significance α = 0.10. If the P-value is less than α, reject the null hypothesis H₀; otherwise, fail to reject H₀.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
P-value
The P-value is a statistical measure that helps determine the significance of results in hypothesis testing. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. A smaller P-value indicates stronger evidence against the null hypothesis, leading researchers to consider rejecting it.
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06:50
Step 3: Get P-Value
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to calculate a test statistic. The outcome determines whether there is enough evidence to reject H0 in favor of H1 at a specified significance level (α).
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06:21Step 1: Write Hypotheses
Significance Level (α)
The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis in hypothesis testing. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common values for α are 0.05, 0.01, and 0.10, with lower values indicating a stricter criterion for significance.
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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
[APPLET] A researcher claims that the mean age of the residents of a small town is more than 38 years. The ages (in years) of a random sample of 30 residents are listed below. At α=0.10, is there enough evidence to support the researcher’s claim? Assume the population standard deviation is 9 years.
Textbook Question
You want your test to support a positive claim about your college, not just fail to reject one. Should you state your claim so that the null hypothesis contains the claim or the alternate hypothesis contains the claim? Explain.
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
Textbook Question
A travel analyst says that the mean price of a meal for a family of 4 in a resort restaurant is at most \$100. A random sample of 33 meal prices for families of 4 has a mean of \$110 and a standard deviation of \$19. At α=0.01, is there enough evidence to reject the analyst’s claim?