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Chapter 8: Problem 55

If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? Why or why not?

Short Answer

Expert verified
No, rejecting the null hypothesis doesn't directly prove it to be false. It just suggests that there's strong evidence against it and that the alternative hypothesis is more likely to be true, considering the observed data and the chosen significance level.

Step by step solution

01

Understanding Null Hypothesis

The null hypothesis, often denoted as H0, is a statement about a population parameter. Typically, it suggests no effect or no difference. While testing the null hypothesis, we assume it to be true until statistical evidence suggests otherwise.
02

Implication of Rejecting the Null Hypothesis

Rejecting the null hypothesis helps us conclude that the observed data is inconsistent with the assumption made under the null hypothesis, often with a significance level considered. Thus, there is enough evidence to believe that the effect or difference is likely to exist in the population.
03

Testing versus Proving

However, rejecting the null hypothesis doesn't directly prove it's false. In statistics, hypothesis testing relies on probability and decision-making under uncertainty. There is always a chance that the null hypothesis is rejected incorrectly due to sampling variability (that's a Type I error). Therefore, instead of 'proving the null hypothesis wrong', we say 'the data provides strong evidence against the null hypothesis.'

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Most popular questions from this chapter

An immunologist is testing the hypothesis that the current flu vaccine is less than \(73 \%\) effective against the flu virus. The immunologist is using a \(1 \%\) significance level and these hypotheses: \(\mathrm{H}_{\mathrm{o}}: p=0.73\) and \(\mathrm{H}_{\mathrm{a}}: p<0.73\). Explain what the \(1 \%\) significance level means in context.

A true/false test has 50 questions. Suppose a passing grade is 35 or more correct answers. Test the claim that a student knows more than half of the answers and is not just guessing. Assume the student gets 35 answers correct out of \(50 .\) Use a significance level of \(0.05 .\) Steps 1 and 2 of a hypothesis test procedure are given. Show steps 3 and 4, and be sure to write a clear conclusion. Step $$\text { 1: } \begin{aligned}&\mathrm{H}_{0}: p=0.50 \\\&\mathrm{H}_{\mathrm{a}}: p>0.50\end{aligned}$$ Step 2: Choose the one-proportion \(z\) -test. Sample size is large enough, because \(n p_{0}\) is \(50(0.5)=25\) and \(n\left(1-p_{0}\right)=50(0.50)=25\), and both are more than \(10 .\) Assume the sample is random and \(\alpha=0.05\).

Suppose you are testing someone to see whether he or she can tell butter from margarine when it is spread on toast. You use many bite-sized pieces selected randomly, half from buttered toast and half from toast with margarine. The taster is blindfolded. The null hypothesis is that the taster is just guessing and should get about half right. When you reject the null hypothesis when it is actually true, that is often called the first kind of error. The second kind of error is when the null is false and you fail to reject. Report the first kind of error and the second kind of error.

Give the null and alternative hypotheses for each test, and state whether a one-proportion z-test or a two-proportion z-test would be appropriate. a. You test a person to see whether he can tell tap water from bottled water. You give him 20 sips selected randomly (half from tap water and half from bottled water) and record the proportion he gets correct to test the hypothesis. b. You test a random sample of students at your college who stand on one foot with their eyes closed and determine who can stand for at least 10 seconds, comparing athletes and nonathletes.

Suppose you are testing someone to see whether she or he can tell Coke from Pepsi, and you are using 20 trials, half with Coke and half with Pepsi. The null hypothesis is that the person is guessing. a. About how many should you expect the person to get right under the null hypothesis that the person is guessing? b. Suppose person A gets 13 right out of 20 , and person B gets 18 right out of 20 . Which will have a smaller \(\mathrm{p}\) -value, and why?
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