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Chapter 8: Problem 2

What is meant by a type I error? A type II error? How are they related?

Short Answer

Expert verified
Type I error is a false positive, Type II error is a false negative. They are inversely related.

Step by step solution

01

Understanding Type I Error

A Type I error occurs when we incorrectly reject the null hypothesis when it is actually true. In other words, it's the false positive rate. When we say something is significant or there is an effect when in fact there is none, we commit a Type I error. It's denoted by the Greek letter alpha (伪) and is often set at 0.05, representing a 5% risk of concluding there is an effect when there isn't.
02

Understanding Type II Error

A Type II error happens when we fail to reject the null hypothesis when it is false and there actually is an effect or difference. This is also known as a false negative. The probability of a Type II error is denoted by beta (尾). Unlike alpha, beta is not commonly predetermined, but smaller values indicate stronger tests.
03

Relation Between Type I and Type II Errors

Type I and Type II errors are related in the context of hypothesis testing and typically involve a trade-off. If you decrease the probability of committing a Type I error (for example, by choosing a smaller alpha), you often increase the probability of committing a Type II error unless you increase the sample size. Optimizing tests involves balancing these two risks while also considering the power of the test, which is 1 minus the Type II error rate (1 - 尾).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error
Imagine you're celebrating a result, claiming there's a significant finding, only to discover it's all a mistake! This might happen due to a Type I error.
In hypothesis testing, a Type I error occurs when we wrongly reject the null hypothesis, which is actually true. It's like crying wolf when there's no wolf.
  • Example: A medical test indicates you have a disease when you don't. That's a false positive.
  • Denoted as alpha (伪), it's often set at 0.05, meaning there's a 5% chance of making this error.
Understanding Type I errors helps ensure that the results claimed are truly significant and not just random noise.
Type II Error
While a Type I error is recognizing a pattern that doesn't exist, a Type II error is missing a pattern that is present, akin to overlooking a real issue.
A Type II error happens when we fail to reject the null hypothesis, even though it's false. It's the false negative that tells us nothing's significant when there's actually something to find.
  • Example: A test fails to detect a disease when in fact the person is ill.
  • Illustrated by beta (尾), its value isn't predetermined like alpha, but smaller bets denote more powerful tests.
Identifying and reducing Type II errors can be crucial, especially when there's potential harm in missing actual evidence.
Null Hypothesis
The starting point of hypothesis testing is the null hypothesis, the default assumption that there's no effect or difference.
We test this assumption to determine if there's enough evidence to reject it, suggesting some phenomenon or effect is present.
  • The purpose is to provide a baseline that can be scientifically challenged.
  • If data renders the null hypothesis implausible, researchers proceed with an alternative hypothesis that suggests a significant effect.
Understanding the null hypothesis acts as a foundation in hypothesis testing, encouraging objective evaluation of data before claiming new discoveries.
Statistical Significance
When someone says that a finding is 'statistically significant,' it means the result isn't likely due to random chance.
Statistical significance helps in determining whether to reject the null hypothesis. It's essential to assess if the effect size is substantial enough to matter or if it's merely a fluke.
  • Typically evaluated with a p-value, where a small value (e.g., less than 0.05) indicates strong evidence against the null hypothesis.
  • While significance suggests reliability, importance is contingent on the real-world applicability of the findings.
By establishing statistical significance, researchers can confidently communicate their findings, knowing that they're meaningful and reliable.

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Most popular questions from this chapter

Define null and alternative hypotheses, and give an example of each.

For Exercises 5 through \(20,\) assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified. Calories in Pancake Syrup A nutritionist claims that the standard deviation of the number of calories in 1 tablespoon of the major brands of pancake syrup is \(60 .\) A random sample of major brands of syrup is selected, and the number of calories is shown. At \(\alpha=0.10,\) can the claim be rejected? $$ \begin{array}{cccccc}{53} & {210} & {100} & {200} & {100} & {220} \\ {210} & {100} & {240} & {200} & {100} & {210} \\ {100} & {210} & {100} & {210} & {100} & {60}\end{array} $$

For Exercises 7 through \(23,\) perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Find the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Assume that the population is approximately normally distributed. Water Consumption The Old Farmer's Almanac stated that the average consumption of water per person per day was 123 gallons. To test the hypothesis that this figure may no longer be true, a researcher randomly selected 16 people and found that they used on average 119 gallons per day and \(s=5.3 .\) At \(\alpha=0.05,\) is there enough evidence to say that the Old Farmer's Almanac figure might no longer be correct? Use the \(P\) -value method.

For each conjecture, state the null and alternative hypotheses. a. The average weight of dogs is 15.6 pounds. b. The average distance a person lives away from a toxic waste site is greater than 10.8 miles. c. The average farm size in 1970 was less than 390 acres. d. The average number of miles a vehicle is driven per year is \(12,603\). e. The average amount of money a person keeps in his or her checking account is less than \(\$ 24\).

IRS Audits The IRS examined approximately $$1 \%$$ of individual tax returns for a specific year, and the average recommended additional tax per return was $$\$ 19,150 .$$ Based on a random sample of 50 returns, the mean additional tax was $$\$ 17,020 .$$ If the population standard deviation is $$\$ 4080,$$ is there sufficient evidence to conclude that the mean differs from $$\$ 19,150$$ at $$\alpha=0.05 ?\( Does a \)95 \%$$ confidence interval support this result?
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