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Chapter 9: Problem 7

Consider \(H_{0}=\mu=20\) versus \(H_{1}: \mu<20 .\) A. What type of error would you make if the null hypothesis is actually false and you fail to reject it? b. What type of error would you make if the null hypothesis is actually true and you reject it?

Short Answer

Expert verified
a) If the null hypothesis is actually false and you fail to reject it, you are committing a Type II error. b) If the null hypothesis is actually true and you reject it, you are committing a Type I error.

Step by step solution

01

Identify Type I Error

A Type I error occurs when we reject a true null hypothesis (\(H_{0}\)), in other words, when we believe something is effective or significant, but it's not. One could say that it's a false positive. In this case, it means if \( \mu = 20 \) is true and we reject it, we make a Type I error.
02

Identify Type II Error

A Type II error happens when the null hypothesis is false and we fail to reject it. Or said another way, the alternative hypothesis is true but we believe the null hypothesis. This can be considered a false negative. In given problem, it implies that if \( \mu < 20 \) is true (meaning the null hypothesis is false), and we fail to reject \( \mu = 20 \) (do not accept \( \mu < 20 \)), we make a Type II error.

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Key Concepts

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

Type I Error
In hypothesis testing, a Type I error occurs when the null hypothesis (H_{0}) is true, but we mistakenly reject it. It's like sounding an alarm when there is no fire. This is considered a **false positive** because it suggests a significant effect exists when it doesn't. In our exercise, if we incorrectly decide that the mean \(\mu<20\) when it's actually \(\mu=20\), we commit a Type I error.

**Key Points**:
  • Type I error can lead to believing a treatment or change is effective when it isn't.
  • The probability of making a Type I error is denoted by \(\alpha\), often set at 0.05.
  • Hence, if our test result supports rejecting the null hypothesis, we do so while accepting there's a small risk of a Type I error.
Type II Error
A Type II error arises when the null hypothesis (H_{0}) is false, yet it is not rejected. Imagine a smoke detector failing to detect a real fire; that's what a **false negative** is like. In the scenario provided, this means \(\mu<20\) is true (the alternative hypothesis), but we incorrectly retain \(\mu=20\), missing the actual effect.

**Important Factors**:
  • Type II error means missing a real effect or change and continues believing an ineffective status quo is correct.
  • Denoted by \(\beta\), the probability of a Type II error is influenced by sample size, effect size, and significance level.
  • It's important to balance Type I and Type II error risks to draw accurate conclusions from your test.
Null Hypothesis
The null hypothesis, denoted as (H_{0}), is a core concept in statistics used to state that there is no effect or no difference. It's what the researcher seeks to test and possibly reject. In the exercise, \(H_{0}: \mu=20\) suggests that the average is 20, which is the status quo or the default assumption.

**Understanding Null Hypothesis**:
  • It serves as a default or starting statement for statistical observation, assertion of "nothing new".
  • Rejecting the null hypothesis implies that there is enough statistical evidence to support an alternative hypothesis (H_{1}).
  • Retaining the null doesn't prove H_{0} is true but rather that there's insufficient evidence to accept a different claim.

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Most popular questions from this chapter

Consider the following null and alternative hypotheses: $$ H_{0}=\mu=60 \text { versus } H_{1}: \mu>60 $$ Suppose you perform this test at \(\alpha=.01\) and fail to reject the null hypothesis. Would you state that the difference between the hypothesized value of the population mean and the observed value of the sample mean is "statistically significant" or would you state that this difference is "statistically not significant"? Explain.

What are the four possible outcomes for a test of hypothesis? Show these outcomes by writing a table. Briefly describe the Type I and Type II errors.

Write the null and alternative hypotheses for each of the following examples. Determine if each is a case of a two-tailed, a left-tailed, or a right-tailed test. a. To test if the mean number of hours spent working per week by college students who hold jobs is different from 20 hours b. To test whether or not a bank's ATM is out of service for an average of more than 10 hours per month c. To test if the mean length of experience of airport security guards is different from 3 years d. To test if the mean credit card debt of college seniors is less than \(\$ 1000\) e. To test if the mean time a customer has to wait on the phone to speak to a representative of a mailorder company about unsatisfactory service is more than 12 minutes

According to a Pew Research Center nationwide telephone survey of American adults conducted by phone between March 15 and April 24, \(2011,75 \%\) of adults said that college education has become too expensive for most people and they cannot afford it (Time, May 30,2011 ). Suppose that this result is true for the 2011 population of American adults. In a recent poll of 1600 American adults, 1160 said that college education has become too expensive for most people and they cannot afford it. Using a \(1 \%\) significance level, perform a test of hypothesis to determine whether the current percentage of American adults who will say that college education has become too expensive for most people and they cannot afford is lower than \(75 \%\). Use both the \(p\) -value and the critical-value approache:

A random sample of 8 observations taken from a population that is normally distributed produced a sample mean of \(44.98\) and a standard deviation of \(6.77 .\) Find the critical and observed values of \(t\) and the ranges for the \(p\) -value for each of the following tests of hypotheses, using \(\alpha=.05\). a. \(H_{0}: \mu=50\) versus \(H_{1}: \mu \neq 50\) b. \(H_{0}: \mu=50\) versus \(H_{1}: \mu<50\)
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