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Chapter 19: Problem 1

Which of the following are true? If false, explain briefly. a. AP-value of 0.01 means that the null hypothesis is false. b. AP-value of 0.01 means that the null hypothesis has a 0.01 chance of being true. c. AP-value of 0.01 is evidence against the null hypothesis. d. AP-value of 0.01 means we should definitely reject the null hypothesis.

Short Answer

Expert verified
a. False, the p-value is just a degree of inconsistency. \n b. False, the p-value is not equivalent to the probability of the null hypothesis being true. \n c. True, a low p-value provides evidence against the null hypothesis. \n d. False, the decision to reject the null hypothesis also depends on the significance level and context.

Step by step solution

01

Answer for Choice a

The statement 'A p-value of 0.01 means that the null hypothesis is false.' is False. The p-value is a degree of inconsistency with the null hypothesis, not a proof that the null hypothesis is false.
02

Answer for Choice b

The statement 'A p-value of 0.01 means that the null hypothesis has a 0.01 chance of being true.' is False. The p-value is not the probability that the null hypothesis is true; instead, it's the probability of obtaining the observed data or data more extreme, given that the null hypothesis is true.
03

Answer for Choice c

The statement 'A p-value of 0.01 is evidence against the null hypothesis.' is True. A p-value below a chosen significance level (like 0.05 or 0.01) generally provides evidence against the null hypothesis.
04

Answer for Choice d

The statement 'A p-value of 0.01 means we should definitely reject the null hypothesis.' is False. While a low p-value provides evidence against the null hypothesis, the decision to reject it also depends on the significance level, the context of the question, and often involves other considerations.

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Key Concepts

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

P-value interpretation
Understanding the interpretation of a p-value is crucial in statistical hypothesis testing. The p-value is a measure of the strength of evidence against the null hypothesis. It represents the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true.

For example, a p-value of 0.01 indicates there is a 1% chance of observing results as extreme as the current data if the null hypothesis is true. Importantly, it does not imply that there is a 1% probability that the null hypothesis is true or false.
  • P-values help us decide whether the observed data significantly contradict what we'd expect under the null hypothesis.
  • A small p-value, typically less than the chosen significance level, suggests that the data strongly contradict the null hypothesis.
  • However, a p-value alone should not be used as definitive proof or disproof of the null hypothesis.
Null Hypothesis
The null hypothesis (\(H_0\)) is a fundamental concept in hypothesis testing. It is a statement that there is no effect or no difference, essentially serving as a default or starting assumption. We test this hypothesis to determine if there is sufficient evidence to support the presence of an effect or difference.

  • The null hypothesis is not a claim that something is true but a benchmark to compare against.
  • When performing statistical tests, we look to see if the data provide strong enough evidence to reject the null hypothesis.
  • The outcome of hypothesis tests often includes a decision to reject or not reject \(H_0\), but never to accept it outright.
In practice, rejecting the null hypothesis suggests sufficient statistical evidence exists for the alternative hypothesis, which posits some effect or difference exists.
Significance Level
The significance level, often denoted by alpha (\(\alpha\)), is a threshold set by the researcher to determine when to reject the null hypothesis. It quantifies the risk of making a Type I error, which occurs when a true null hypothesis is incorrectly rejected.

For instance, a commonly used significance level is 0.05, which means there is a 5% risk of rejecting the null hypothesis when it is actually true.
  • If the p-value is less than or equal to the significance level, we reject the null hypothesis, suggesting the results are statistically significant.
  • A lower significance level (like 0.01) indicates a more stringent criterion for rejecting the null hypothesis.
  • The choice of significance level can depend on the field of study, the context of the decision, and the consequences of making an error.
Choosing an appropriate significance level is an important decision that impacts the reliability and interpretation of statistical tests.
Statistical Evidence
Collecting and interpreting statistical evidence is the heart of hypothesis testing. Statistical evidence involves using data to determine the likelihood of a hypothesis being true or false. It is not absolute proof but a measure of the likelihood that our observations are consistent with what the hypothesis claims.

  • P-values provide a measure of statistical evidence against the null hypothesis.
  • The strength of statistical evidence is often assessed in conjunction with the chosen significance level.
  • Decisions based on statistical evidence should account for the context, including possible errors and the broader implications of the results.
To make informed decisions, it is crucial to understand not only the numerical outcomes of statistical tests but also their implications and limitations. Statistical evidence, therefore, supports scientific claims and guides decision-making processes across various fields.

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Most popular questions from this chapter

A medical researcher tested a new treatment for poison ivy against the traditional ointment. He concluded that the new treatment is more effective. Explain what the P-value of 0.047 means in this context.

Have harsher penalties and ad campaigns increased seat-belt use among drivers and passengers? Observations of commuter traffic failed to find evidence of a significant change compared with three years ago. Explain what the study's P-value of 0.17 means in this context.

In 2015 , the U.S. Census Bureau reported that \(62.2 \%\) of American families owned their homes the lowest rate in 20 years. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers to encourage people to become homeowners. They decide to adopt the plan on a 2 -year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. a. In words, what will their hypotheses be? b. What would a Type I error be? c. What would a Type II error be? d. For each type of error, tell who would be harmed. e. What would the power of the test represent in this context?

A new reading program may reduce the number of elementary school students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale test involving nearly 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not meet the grade- level goal was reduced from \(15.9 \%\) to \(15.1 \%\). The hypothesis that the new reading program produced no improvement was rejected with a P-value of 0.023 . a. Explain what the P-value means in this context. b. Even though this reading method has been shown to be significantly better, why might you not recommend that your local school adopt it?

An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About \(40 \%\) break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay from another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. a. Suppose the new, expensive clay really is no better than her usual clay. What's the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.) b. If she decides to switch to the new clay and it is no better, what kind of error did she commit? c. If the new clay really can reduce breakage to only \(20 \%,\) what's the probability that her test will not detect the improvement? d. How can she improve the power of her test? Offer at least two suggestions.
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