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Chapter 8: Problem 56

If we do not reject the null hypothesis, is it valid to say that we accept the null hypothesis? Why or why not?

Short Answer

Expert verified
No, it's not valid to say that we accept the null hypothesis if we do not reject it. The correct interpretation is that there is not enough evidence to reject the null hypothesis, not that the null hypothesis is accepted or proven true.

Step by step solution

01

Concept Understanding

In hypothesis testing, when we say that we do not reject the null hypothesis, it does not imply that we entirely accept the null hypothesis. Instead, it means that there's not enough evidence in our data to conclude that the null hypothesis is false, beyond a reasonable doubt.
02

Reasoning

The reason we don't say that we 'accept' the null hypothesis is because there might always be some effect due to random chance alone, even if it's very small. Even if the data suggest that the null hypothesis might be true, there's always a chance that the observed effect was due to random chance. Thus, we say 'fail to reject' rather than 'accept'.
03

Example Illustration

Consider a courtroom analogy. Here, a defendant is 'innocent until proven guilty'. This doesn't mean that they're declared innocent if they're not found guilty. It merely means that there wasn't enough evidence to convict them. Similarly, failing to reject the null hypothesis does not prove it to be true, it merely means there wasn't sufficient evidence against it.

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Most popular questions from this chapter

Refer to Exercise 8.3. Suppose 100 people attend boot camp and 44 of them return to prison within three years). The population recidivism rate for the whole state is \(40 \%\). a. What is \(\hat{p}\), the sample proportion of successes? (It is somewhat odd to call retuming to prison a success.) b. What is \(p_{0}\), the hypothetical proportion of success under the null hypothesis? c. What is the value of the test statistic? Explain in context.

Feder and Dugan (2002) reported a study in which 404 domestic violence defendants were randomly assigned to counseling and probation (the experimental group) or just probation (the control group). Out of 230 people in the counseling group, 55 were arrested within 12 months. Out of 174 people assigned to probation, 42 were arrested within 12 months. Determine whether counseling lowered the arrest rate; use a \(0.05\) significance level. Start by comparing the percentages.

A teacher giving a true/false test wants to make sure her students do better than they would if they were simply guessing, so she forms a hypothesis to test this. Her null hypothesis is that a student will get \(50 \%\) of the questions on the exam correct. The alternative hypothesis is that the student is not guessing and should get more than \(50 \%\) in the long run. $$ \begin{aligned} &\mathrm{H}_{0}=p=0.50 \\ &\mathrm{H}_{\mathrm{a}}: p>0.50 \end{aligned} $$ A student gets 30 out of 50 questions, or \(60 \%\), correct. The p-value is \(0.079 .\) Explain the meaning of the p-value in the context of this question.

An arthritis diet claims that the disease can be relieved by reducing sugar from the diet. To test this claim, a researcher randomly assigns arthritis patients to two groups. Both groups eat the same amount of calories, but one group eats almost no sugar and the other group includes sugar in their meal. After 3 months, the doctor tests the claim that the sugar-free diet is better than the usual diet. She records the proportion of each group that got relieved of almost \(10 \%\) of their problem. She then announced that she failed to reject the null hypothesis. Which of the following are valid interpretations of her findings? a. The sugar-free dict was less effective than the normal diet. b. The researcher did not see enough evidence to conclude that the sugarfree diet was more effective. c. The sugar-free diet and the normal diet were equally effective. d. There were no significant differences in effectiveness between the sugar- free diet and normal diet.

Is it acceptable practice to look at your research results, note the direction of the difference, and then make the alternative hypothesis one-sided in order to achieve a significant difference? Explain.
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