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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 random sample of 500 observations produced a sample proportion equal to \(.38 .\) Find the critical and observed values of \(z\) for each of the following tests of hypotheses using \(\alpha=.05\). a. \(H_{0}: p=.30\) versus \(H_{1}: p>.30\) b. \(H_{0}: p=.30\) versus \(H_{1}: p \neq .30\)

Consider \(H_{0}=p=.45\) versus \(H_{1}: p<.45\). a. A random sample of 400 observations produced a sample proportion equal to .42. Using \(\alpha=.025\), would you reject the null hypothesis? b. Another random sample of 400 observations taken from the same population produced a sample proportion of \(.39 .\) Using \(\alpha=.025\), would you reject the null hypothesis? Comment on the results of parts a and \(\mathrm{b}\).

A soft-drink manufacturer claims that its 12 -ounce cans do not contain, on average, more than 30 calories. A random sample of 64 cans of this soft drink, which were checked for calories, contained a mean of 32 calories with a standard deviation of 3 calories. Does the sample information support the altemative hypothesis that the manufacturer's claim is false? Use a significance level of \(5 \%\). Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value and \(\alpha=.05 ?\)

According to a book published in \(2011,45 \%\) of the undergraduate students in the United States show almost no gain in learning in their first 2 years of college (Richard Arum et al., Academically Adrift, University of Chicago Press, Chicago, 2011 ). A recent sample of 1500 undergraduate students showed that this percentage is \(38 \%\). Can you reject the null hypothesis at a \(1 \%\) significance level in favor of the alternative that the percentage of undergraduate students in the United States who show almost no gain in learning in their first 2 years of college is currently lower than \(45 \%\) ? Use both the \(p\) -value and the critical-value approaches.

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.
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