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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

In a 2018 study reported in The Lancet, Mercie et al. investigated the efficacy and safety of varenicline for smoking cessation in people living with HIV. The study was a randomized, double-blind, placebo-controlled trial. Of the 123 subjects treated with varenicline, 18 abstained from smoking for the entire 48-week study period. Of the 124 subjects assigned to the placebo group, 8 abstained from smoking for the entire study period. a. Find the sample percentage of subjects in each group who abstained from smoking for the entire study period. b. Determine whether varenicline is effective in reducing smoking among HIV patients. Note that this means we should test if the proportion of the varenicline group who abstained from smoking for the entire study period is significantly greater than that of the placebo group. Use a significance level of \(0.05\)

For each of the following, state whether a one-proportion \(z\) -test or a two- proportion \(z\) -test would be appropriate, and name the population(s). a. A polling agency takes a random sample of voters in California to determine if a ballot proposition will pass. b. A researcher asks a random sample of residents from coastal states and a random sample of residents of non-coastal states whether they favor increased offshore oil drilling. The researcher wants to determine if there is a difference in the proportion of residents who support off-shore drilling in the two regions.

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

In a 2018 study reported in The Lancet, Molina et al. reported on a study for treatment of patients with HIV-1. The study was a randomized, controlled, double-blind study that compared the effectiveness of ritonavir-boosted darunavir (rbd), the drug currently used to treat HIV-1, with dorovirine, a newly developed drug. Of the 382 subjects taking ritonavir-boosted darunavir, 306 achieved a positive result. Of the 382 subjects taking dorovirine, 321 achieved a positive outcome. See page 430 for guidance. a. Find the sample percentage of subjects who achieved a positive outcome in each group. b. Perform a hypothesis test to test whether the proportion of patients who achieve a positive outcome with the current treatment (ritonavir-boosted darunavir) is different from the proportion of patients who achieve a positive outcome with the new treatment (dorovirine). Use a significance level of \(0.01\). Based on this study, do you think dorovirine might be a more effective treatment option for HIV-1 than ritonavir-boosted darunavir? Why or why not?

A 2003 study of dreaming published in the journal Perceptual and Motor Skills found that out of a random sample of 113 people, 92 reported dreaming in color. However, the proportion of people who reported dreaming in color that was established in the 1940 s was \(0.29\) (Schwitzgebel 2003). Check to see whether the conditions for using a one-proportion z-test are met assuming the researcher wanted to see whether the proportion dreaming in color had changed since the \(1940 \mathrm{~s}\).
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