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Chapter 8: Problem 7

What symbols are used to represent the probabilities of type I and type II errors?

Short Answer

Expert verified
Type I error is represented by \( \alpha \), and Type II error by \( \beta \).

Step by step solution

01

Understanding Type I Error

A Type I error occurs when we reject a true null hypothesis. The probability of a Type I error is represented by the Greek letter \( \alpha \). It is also known as the significance level of a test.
02

Understanding Type II Error

A Type II error occurs when we fail to reject a false null hypothesis. The probability of making a Type II error is represented by the Greek letter \( \beta \).

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Key Concepts

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

Type I Error
A Type I error is a statistical mistake, sometimes referred to as a 'false positive'. It happens when we incorrectly reject a true null hypothesis. Imagine you are testing a new drug, and you mistakenly conclude that it works when it actually doesn't. This is a Type I error. The probability of making this kind of error is denoted by the Greek letter \( \alpha \).

In practical terms, setting \( \alpha \) means deciding how willing you are to make this mistake. If you choose a small \( \alpha \), like 0.01, you're being very cautious. It means you're accepting only a 1% chance of wrongly rejecting the null hypothesis.
  • If \( \alpha = 0.05\), this means there's a 5% risk of concluding that there's an effect when there actually isn't.
  • Always balance the risk of Type I error with experimentation needs; smaller \( \alpha \) usually requires larger sample sizes.
Type II Error
Type II error occurs when one fails to reject a false null hypothesis. It's like a 'false negative' in medical testing. For instance, you might test a drug and believe it has no effect when it actually does. The probability of making a Type II error is represented by the Greek letter \( \beta \).

The value of \( \beta \) depends on several factors:
  • The true situation, which is unknown, affects \( \beta \) very much.
  • Sample size: Larger samples can lower \( \beta \) because they provide more information.
  • The chosen significance level (\( \alpha \)); a smaller \( \alpha \) may increase the chance of \( \beta \).
Understanding and acknowledging the risk of Type II errors helps scientists design better experiments and ensure more reliable conclusions. The power of a test, calculated as \( 1 - \beta \), indicates its sensitivity to detect a true effect when there is one.
Significance Level
Significance level is a crucial aspect of hypothesis testing. It is the threshold set by researchers to determine whether to reject the null hypothesis. Represented by \( \alpha \), it indicates the probability of making a Type I error.

The choice of \( \alpha \) reflects the balance between caution and efficiency in testing, which directly influences the reliability and validity of results.
  • A common choice for \( \alpha \) in many fields is 0.05, meaning a 5% chance of falsely rejecting the null hypothesis.
  • In more rigorous fields, like medicine, a lower \( \alpha \) such as 0.01 might be used to minimize risks.
Setting the right significance level depends on the context of the research and the potential consequences of errors involved. Lower significance levels demand bigger sample sizes, which can be resource-intensive, yet they provide more confidence in the results.

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Most popular questions from this chapter

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. Heights of Tall Buildings A researcher estimates that the average height of the buildings of 30 or more stories in a large city is at least 700 feet. A random sample of 10 buildings is selected, and the heights in feet are shown. At \(\alpha=0.025,\) is there enough evidence to reject the claim? $$ \begin{array}{ccccc}{485} & {511} & {841} & {725} & {615} \\ {520} & {535} & {635} & {616} & {582}\end{array} $$

For Exercises 5 through \(20,\) perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Natural Gas Heat The Energy Information Administration reported that \(51.7 \%\) of homes in the United States were heated by natural gas. A random sample of 200 homes found that 115 were heated by natural gas. Does the evidence support the claim, or has the percent- age changed? Use \(\alpha=0.05\) and the \(P\) -value method. What could be different if the sample were taken in a different geographic area?

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. Soda Bottle Content A machine fills 12 -ounce bottles with soda. For the machine to function properly, the standard deviation of the population must be less than or equal to 0.03 ounce. A random sample of 8 bottles is selected, and the number of ounces of soda in each bottle is given. At \(\alpha=0.05,\) can we reject the claim that the machine is functioning properly? Use the \(P\) -value method. $$ \begin{array}{cccc}{12.03} & {12.10} & {12.02} & {11.98} \\ {12.00} & {12.05} & {11.97} & {11.99}\end{array} $$

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

For Exercises I through 25, perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use diagrams to show the critical region (or regions), and use the traditional method of hypothesis testing unless otherwise specified. Speeding Tickets A motorist claims that the South Boro Police issue an average of 60 speeding tickets per day. These data show the number of speeding tickets issued each day for a randomly selected period of 30 days. Assume \(\sigma\) is \(13.42 .\) Is there enough evidence to reject the motorist's claim at \(\alpha=0.05 ?\) Use the \(P\) -value method. $$ \begin{array}{cccccccc}{72} & {45} & {36} & {68} & {69} & {71} & {57} & {60} \\\ {83} & {26} & {60} & {72} & {58} & {87} & {48} & {59} \\ {60} & {56} & {64} & {68} & {42} & {57} & {57} \\ {58} & {63} & {49} & {73} & {75} & {42} & {63}\end{array} $$
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