Textbook Question
Type I Error and Type II Error
a. In general, what is a type I error? In general, what is a type II error?
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9. Hypothesis Testing for One Sample
Type I & Type II Errors
2:02 minutesProblem 10.1.16
Textbook Question
In Problems 13–20, (a) state the null and alternative hypotheses in words, (b) state the null and alternative hypotheses symbolically, (c) explain what it would mean to make a Type I error, and (d) explain what it would mean to make a Type II error.
Fair Packaging and Labeling Federal law requires that a jar of peanut butter that is labeled as containing 32 ounces must contain at least 32 ounces. A consumer advocate feels that a certain peanut butter manufacturer is shorting customers by underfilling the jars.
Verified step by step guidance1
Step 1: Identify the parameter of interest and the claim. Here, the parameter is the true amount of peanut butter in a jar, and the claim is that the jars contain at least 32 ounces as labeled.
Step 2: State the null hypothesis (H0) and alternative hypothesis (Ha) in words. The null hypothesis represents the status quo or the claim to be tested, and the alternative represents the consumer advocate's suspicion.
Step 3: Write the hypotheses symbolically. Since the law requires at least 32 ounces, the null hypothesis will be that the mean amount \( \mu \) is greater than or equal to 32 ounces, and the alternative will be that \( \mu \) is less than 32 ounces.
Step 4: Explain what a Type I error means in this context. A Type I error occurs if we reject the null hypothesis when it is actually true — that is, concluding the jars are underfilled when they actually meet or exceed 32 ounces.
Step 5: Explain what a Type II error means in this context. A Type II error occurs if we fail to reject the null hypothesis when the alternative is true — that is, concluding the jars are not underfilled when they actually contain less than 32 ounces.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hypothesis Testing
Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a presumed statement (null hypothesis) about a population. It involves formulating a null hypothesis (H0) and an alternative hypothesis (Ha), then using sample data to determine which hypothesis is supported.
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Performing Hypothesis Tests: Proportions
Type I and Type II Errors
Type I error occurs when the null hypothesis is wrongly rejected, meaning a false positive. Type II error happens when the null hypothesis is wrongly accepted, meaning a false negative. Understanding these errors helps evaluate the risks of incorrect conclusions in hypothesis testing.
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04:24Types of Data
Formulating Hypotheses in Context
Formulating hypotheses requires translating the problem context into statistical statements. For example, the null hypothesis might state the jar contains at least 32 ounces, while the alternative suggests it contains less. Clear hypotheses guide the testing process and interpretation of results.
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01:35Step 1: Write Hypotheses Example 1
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