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Chapter 14: Problem 17

Someone claims that, given a \(p\) -value less than .01, the corresponding null hypothesis also must be true with probability less than .01. Comments?

Short Answer

Expert verified
No, the claim is incorrect. A p-value less than .01 does not denote that the null hypothesis is true with a probability of less than .01. It represents the chance of obtaining the observed data or more extreme, assuming the null hypothesis is true, is less than .01.

Step by step solution

01

Understanding the P-value

The first step is realizing that the p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic assuming the null hypothesis is true.
02

Relationship between P-value and Hypothesis

Despite commonly held misconceptions, the p-value is not the probability that the null hypothesis is true. Instead, it quantifies how evidence opposes the null hypothesis.
03

Conclusion

So, a p-value less than .01 does not imply that the null hypothesis is true with probability less than .01. It means that, given the null hypothesis is true, the chance of observing the data is less than .01.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Null Hypothesis
The null hypothesis is a foundational element of statistical testing. It is a general statement or default position that there is no effect or no difference in the context of the experiment or study. Imagine you want to test if a new drug works better than a placebo. The null hypothesis would state that there is no difference in outcomes between the drug and the placebo.

In hypothesis testing, researchers aim to evaluate the strength of evidence against the null hypothesis. By initially assuming that the null hypothesis is true, they can then assess whether the data collected supports this assumption or suggests alternative outcomes.
  • The null hypothesis is often denoted as \(H_0\).
  • If evidence is strong enough to reject \(H_0\), an alternative hypothesis \(H_a\) is considered.
Understanding the null hypothesis helps to clarify the approach and objectives of statistical analysis, allowing for clearer interpretation of results.
Interpreting Statistical Significance
Statistical significance helps determine if an observed effect is likely due to chance or reflects a true effect. When researchers say a result is statistically significant, they typically mean that it is unlikely to have occurred by random chance alone, according to a pre-defined threshold.

This threshold is determined by the significance level, often denoted by \( \alpha \). A common choice is \( \alpha = 0.05 \), meaning there's a 5% risk of concluding that an effect exists when it doesn't (false positive). Lower values, like \( \alpha = 0.01 \), indicate stricter criteria.
  • Statistical significance does not imply practical significance.
  • Significance is about the reliability of the result, not its importance.
It's crucial to remember that statistical significance merely informs us of the probability that the null hypothesis would be true under the current data, not the effect's magnitude or importance.
Essentials of Hypothesis Testing
Hypothesis testing is a systematic method used to make inferences or draw conclusions based on sample data. It involves several steps to determine whether there is enough evidence to reject or fail to reject a hypothesis.

At the heart of hypothesis testing lies the comparison between the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). Researchers collect data, calculate relevant statistics, and determine the probability of observing the results under the null hypothesis (this is where the p-value comes into play).
  • A low p-value (< \( \alpha \)) indicates strong evidence against \(H_0\).
  • Failing to reject \(H_0\) does not prove it true, nor does it confirm \(H_a\).
Hypothesis testing helps in making informed decisions based on data. It highlights the strength of evidence needed to make such decisions and is a fundamental method of scientific research and inquiry.

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Most popular questions from this chapter

After testing several thousand high school seniors, a state department of education reported a statistically significant difference between the mean GPAs for female and male students. Comments?

ldentifying the treatment group with \(\mu_{1},\) specify both the null and alternative hypotheses for each of the following studies. Select a directional alternative hypothesis only when a word or phrase justifies an exclusive concern about population mean differences in a particular direction. (a) After randomly assigning migrant children to two groups, a school psychologist determines whether there is a difference in the mean reading scores between groups exposed to either a special bilingual or a traditional reading program. (b) On further reflection, the school psychologist decides that, because of the extra expense of the special bilingual program, the null hypothesis should be rejected only if there is evidence that reading scores are improved, on average, for the group exposed to the special bilingual program. (c) An investigator wishes to determine whether, on average, cigarette consumption is reduced for smokers who chew caffeine gum. Smokers in attendance at an antismoking workshop are randomly assigned to two groups - one that chews caffeine gum and one that does not - and their daily cigarette consumption is monitored for six months after the workshop. (d) A political scientist determines whether males and females differ, on average, about the amount of money that, in their opinion, should be spent by the U.S. government on homeland security. After being informed about the size of the current budget for homeland security, in billions of dollars, randomly selected males and females are asked to indicate the percent by which they would alter this amount- -for example, -8 percent for an 8 percent reduction, 0 percent for no change, 4 percent for a 4 percent increase.

Indicate which member of each of the following pairs of \(p\) -values describes the more rare test result: \(\left(\mathbf{a}_{1}\right) p>.05\) \(\left(\mathbf{a}_{2}\right) \quad p<.05\) \(\left(\mathbf{b}_{1}\right) p<.001\) \(\left(\mathbf{b}_{2}\right) p<.01\) \(\left(\mathbf{c}_{1}\right) p<.05\) \(\left(\mathbf{c}_{2}\right) p<.01\) \(\left(\mathbf{d}_{1}\right) p<.10\) \(\left(\mathbf{d}_{2}\right) \quad p<.20\) \(\left(\mathbf{e}_{1}\right) \quad p=.04\) \(\left(\mathbf{e}_{2}\right) \quad p=.02\)

During recent decades, there have been a series of widely publicized, seemingly transitory, often contradictory research findings reported in newspapers and on television. For example, a few initial research findings suggested that vaccination causes autism in children. However, subsequent, more extensive research findings, as well as a more critical look at the original findings, suggested that vaccination doesn't cause autism (https://www.sciencebasedmedicine.org/reference/vaccinesand-autism/#overview). What might be one explanation for the seemingly erroneous initial research finding?

An investigator wishes to determine whether alcohol consumption causes a deterioration in the performance of automobile drivers. Before the driving test, subjects drink a glass of orange juice, which, in the case of the treatment group, is laced with two ounces of vodka. Performance is measured by the number of errors made on a driving simulator. A total of 120 volunteer subjects are randomly assigned, in equal numbers, to the two groups. For subjects in the treatment group, the mean number of errors \(\left(\bar{X}_{1}\right)\) equals \(26.4,\) and for subjects in the control group, the mean number of errors \(\left(\bar{X}_{2}\right)\) equals 18.6 . The estimated standard error equals \(2.4 .\) (a) Use \(t\) to test the null hypothesis at the .05 level of significance. (b) Specify the \(p\) -value for this test result. (c) If appropriate, construct a 95 percent confidence interval for the true population mean difference and interpret this interval. (d) If the test result is statistically significant, use Cohen's \(d\) to estimate the effect size, given that the standard deviation, \(s_{p}\), equals 13.15 . (e) State how these test results might be reported in the literature, given \(s_{1}=13.99\) and \(s_{2}=12.15\)
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