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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

food company is planning to market a new type of frozen yogurt. However, before marketing thit yogurt, the company wants to find what percentage of the people like it. The company's management has decided that it will market this yogurt only if at least \(35 \%\) of the people like it. The company's researcl department selected a random sample of 400 persons and asked th taste this yogurt. Of these 400 persons, 112 said they liked a. Testing at a \(2.5 \%\) significance level, can you conclude that the company should market this yogurt b. What will your decision be in part a if the probability of making a Type I error is zero? Explain Make the test of part a using the \(p\) -value approach.min

In each of the following cases, do you think the sample size is large enough to use the normal distribution to make a test of hypothesis about the population proportion? Explain why or why not. a. \(n=40\) and \(p=.11\) b. \(n=100\) and \(p=.7\) c. \(n=80 \quad\) and \(\quad p=.05\) d. \(n=50\) and \(p=.14\)

Consider \(H_{0}=\mu=40\) versus \(H_{1}: \mu>40 .\) a. A random sample of 64 observations taken from this population produced a sample mean of 43 and a standard deviation of \(5 .\) Using \(\alpha=.025\), would you reject the null hypothesis? b. Another random sample of 64 observations taken from the same population produced a sample mean of 41 and a standard deviation of \(7 .\) Using \(\alpha=.025\), would you reject the null hypothesis? Comment on the results of parts a and \(\mathrm{b}\).

Explain how the tails of a test depend on the sign in the alternative hypothesis. Describe the signs in the null and alternative hypotheses for a two-tailed, a left-tailed, and a right-tailed test, respectively.

A company claims that the mean net weight of the contents of its All Taste cereal boxes is at least 18 ounces. Suppose you want to test whether or not the claim of the company is true. Explain briefly how you would conduct this test using a large sample. Assume that \(\sigma=.25\) ounce.
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