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Chapter 9: Problem 2

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.

Short Answer

Expert verified
The four potential outcomes in a hypothesis test are: 1) Correct decision when we reject a false null hypothesis, 2) Type I Error when we reject a true null hypothesis, 3) Correct decision when we do not reject a true null hypothesis, 4) Type II Error when we do not reject a false null hypothesis. A Type I Error is the wrongful rejection of a true null hypothesis, while a Type II Error is the failure to reject a false null hypothesis.

Step by step solution

01

Identify The Four Possible Outcomes for a Hypothesis Test

When carrying out a hypothesis test, the four potential results occur from the combination of the possible decisions (rejecting or not rejecting the null hypothesis) with the possible realities (the null hypothesis is true or false). Hence, the four potential outcomes are: 1) Correct decision (reject a false null hypothesis), 2) Type I Error (reject a true null hypothesis), 3) Correct decision (do not reject a true null hypothesis), 4) Type II Error (do not reject a false null hypothesis).
02

Create the Table

A table is a convenient way to display these outcomes. Label the columns as 'Decision' with subcolumns 'Do not Reject H0' and 'Reject H0' (H0: null hypothesis). Label the rows as 'Reality' with subrows 'H0 is true' and 'H0 is false'. The four cells in the table represent the four outcomes already identified.
03

Define Type I and Type II Errors

A Type I Error (False alarm) happens when you reject the null hypothesis when it is, in fact, true. It's like an 'innocent' hypothesis being falsely convicted. On the other hand, a Type II Error (miss) occurs when you fail to reject the null hypothesis when it is false. It's like a 'guilty' hypothesis being exonerated.

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Most popular questions from this chapter

A study claims that all adults spend an average of 14 hours or more on chores during a weekend. A researcher wanted to check if this claim is true. A random sample of 200 adults taken by this researcher showed that these adults spend an average of \(14.65\) hours on chores during a weekend. The population standard deviation is known to be \(3.0\) hours. a. Find the \(p\) -value for the hypothesis test with the alternative hypothesis that all adults spend more than 14 hours on chores during a weekend. Will you reject the null hypothesis at \(\alpha=.01 ?\) b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.01\).

The police that patrol a heavily traveled highway claim that the average driver exceeds the 65 miles per hour speed limit by more than 10 miles per hour. Seventy-two randomly selected cars were clocked by airplane radar. The average speed was \(77.40\) miles per hour, and the standard deviation of the speeds was \(5.90\) miles per hour. Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value range and \(\alpha=.02\) ?

For each of the following examples of tests of hypothesis about the population proportion, show the rejection and nonrejection regions on the graph of the sampling distribution of the sample proportion. a. A two-tailed test with \(\alpha=.05\) b. A left-tailed test with \(\alpha=.02\) c. A right-tailed test with \(\alpha=.025\)

The manager of a service station claims that the mean amount spent on gas by its customers is $$\$ 15.90$$ per visit. You want to test if the mean amount spent on gas at this station is different from $$\$ 15.90$$ per visit. Briefly explain how you would conduct this test when \(\sigma\) is not known. \(9.73\) A tool manufacturing company claims that its top-of-the-line machine that is used to manufacture bolts produces an average of 88 or more bolts per hour. A company that is interested in buying this machine wants to check this claim. Suppose you are asked to conduct this test. Briefly explain how you would do so when \(\sigma\) is not known.

The manager of a restaurant in a large city claims that waiters working in all restaurants in his city eart an average of \(\$ 150\) or more in tips per week. A random sample of 25 waiters selected from restaurants of this city yielded a mean of \(\$ 139\) in tips per week with a standard deviation of \(\$ 28\). Assume that the weekly tips for all waiters in this city have a normal distribution a. Using a \(1 \%\) significance level, can you conclude that the manager's claim is true? Use both approaches b. What is the Type I error in this exercise? Explain. What is the probability of making such an error'
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