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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 problem \(8.15\) the nutritionist was interested in knowing if the rate of vegetarianism in American adults has increased. She carried out a hypothesis test and found that the observed value of the test statistic was \(2.77 .\) We can calculate that the p-value associated with this is \(0.0028\), which is very close to 0\. Explain the meaning of the p-value in this context. Based on this result, should the nutritionist believe the null hypothesis is true?

A proponent of a new proposition on a ballot wants to know whether the proposition is likely to pass. Suppose a poll is taken, and 580 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this question? Explain. If it is a hypothesis test, state the hypotheses and find the test statistic, p-value, and conclusion. If a confidence interval is appropriate, find the approximate \(95 \%\) confidence interval. In both cases, assume that the necessary conditions have been met.

Morse determined that the percentage of \(t\) 's in the English language in the 1800 s was \(9 \%\). A random sample of 600 letters from a current newspaper contained \(48 t\) 's. Using the \(0.10\) level of significance, test the hypothesis that the proportion of \(t\) 's in this modern newspaper is \(0.09\).

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 \(0.05\)

When comparing two sample proportions with a two-sided alternative hypothesis, all other factors being equal, will you get a smaller p-value with a larger sample size or a smaller sample size? Explain.
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