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Chapter 9: Problem 16

Statistical significance Asked to explain the meaning of 鈥渟tatistically significant at the A 0.05 level,鈥� a student says, 鈥淭his means that the probability that the null hypothesis is true is less than 0.05.鈥� Is this explanation correct? Why or why not?

Short Answer

Expert verified
The student's explanation is incorrect; statistical significance at the 0.05 level means observing data unlikely under the null hypothesis, not that its truth probability is less than 0.05.

Step by step solution

01

Understanding Statistical Significance

Statistical significance at the 0.05 level means that in hypothesis testing, the p-value obtained from the data is less than 0.05. The p-value is a measure of the probability of observing the data, or something more extreme, assuming that the null hypothesis is true.
02

Correct Interpretation of p-value

A p-value less than 0.05 suggests that the observed data is unlikely under the null hypothesis. Therefore, if the p-value is below 0.05, we reject the null hypothesis, concluding there is sufficient evidence to suggest the alternative hypothesis may be true. This doesn't mean the probability that the null hypothesis is true is less than 0.05.
03

Common Misconception

The statement provided by the student is a common misconception. The p-value measures the strength of the evidence against the null hypothesis, not the probability that the null hypothesis is true. Thus, saying that the probability that the null hypothesis is true is less than 0.05 is incorrect.
04

Correct Explanation

The correct explanation should be: if a result is statistically significant at the 0.05 level, it means that the probability of observing such a result by chance alone (if the null hypothesis were true) is less than 5%. This gives us a rationale to reject the null hypothesis but does not confirm it is false.

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Key Concepts

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

Null Hypothesis
The null hypothesis is a fundamental concept in statistics and hypothesis testing. It represents a statement or position of no effect or no difference. In an experiment or study, it assumes that any observed effect is purely due to random chance. For example, if we are testing a new drug, the null hypothesis might be that the drug has no effect on patients compared to a placebo.

The null hypothesis is often denoted as \( H_0 \). When conducting a hypothesis test, your goal is often to assess whether the data provides enough evidence to reject this null hypothesis.
  • Example: In a study testing a weight loss supplement, \( H_0 \) would suggest that the supplement does not result in weight loss above what could be expected from random variation or placebo.
  • Role: It serves as a starting point for hypothesis testing, providing a premise to be tested against data.
P-Value
The p-value is a vital component in hypothesis testing. It is a measure that helps determine the significance of your test results. Essentially, it tells you how extreme your data is under the premise that the null hypothesis is true.

Let's break it down:
  • If the p-value is low (typically below 0.05), it suggests that the observed data is unlikely under the null hypothesis, providing evidence against it.
  • Conversely, a high p-value indicates that your data is more consistent with the null hypothesis being true.
The p-value helps you make a decision about the null hypothesis:
  • Low p-value (<0.05): Reject the null hypothesis.
  • High p-value (鈮�0.05): Fail to reject the null hypothesis.
Hypothesis Testing
Hypothesis testing is a statistical method that uses sample data to evaluate a hypothesis about a population parameter. Here is how it works:

  • Start by defining both a null hypothesis (\( H_0 \)) and an alternative hypothesis (\( H_1 \)).
  • Collect data relevant to your hypotheses.
  • Calculate a p-value to determine the strength of evidence against the null hypothesis.
After computing the p-value, you decide whether to reject the null hypothesis.
  • The decision to reject or not is often based on a significance level, typically set at 0.05.
  • If the data yields a p-value less than this level, the null hypothesis is rejected in favor of the alternative hypothesis.

Hypothesis testing helps researchers understand results within the context of chance and variability.
Alternative Hypothesis
While the null hypothesis suggests no effect or no difference, the alternative hypothesis posits the opposite. It is the statement we consider might be true if the null hypothesis is rejected.

The alternative hypothesis is denoted as \( H_1 \) or \( H_a \). Here are some key aspects:
  • Contrasts with the null hypothesis, suggesting a possible effect or difference.
  • Defines the direction or nature of the effect you are testing.
For example, if you are testing a new teaching method, the alternative hypothesis might suggest that this method results in higher student performance compared to traditional methods.

Choosing the alternative hypothesis carefully ensures that the hypothesis test remains objective and relevant to your research questions.

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Most popular questions from this chapter

Heat through the glass How well materials conduct heat matters when designing houses, for example. Conductivity is measured in terms of watts of heat power transmitted per square meter of surface per degree Celsius of temperature difference on the two sides of the material. In these units, glass has conductivity about 1. The National Institute of Standards and Technology provides exact data on properties of materials. Here are measurements of the heat conductivity of 11 randomly selected pieces of a particular type of glass: 1.11 1.07 1.11 1.07 1.12 1.08 1.08 1.18 1.18 1.18 1.12 Is there convincing evidence that the conductivity of this type of glass is greater than 1? Carry out a test to help you answer this question.

Bullies in middle school A University of Illinois study on aggressive behavior surveyed a random sample of 558 middle school students. When asked to describe their behavior in the last 30 days, 445 students said their behavior included physical aggression, social ridicule, teasing, name-calling, and issuing threats. This behavior was not defined as bullying in the questionnaire. Is this evidence that more than three-quarters of the students at that middle school engage in bullying behavior? To find out, Maurice decides to perform a significance test. Unfortunately, he made a few errors along the way. Your job is to spot the mistakes and correct them. $$ \begin{array}{l}{H_{0} : p=0.75} \\ {H_{a} : \hat{p}>0.797}\end{array} $$ where \(p=\) the true mean proportion of middle school students who engaged in bullying. A random sample of 558 middle school students was surveyed. \(558(0.797)=444.73\) is at least 10 $$ z=\frac{0.75-0.797}{\sqrt{\frac{0.797(0.203)}{445}}}=-2.46 ; P \text { -value }=2(0.0069)=0.0138 $$ The probability that the null hypothesis is true is only \(0.0138,\) so we reject \(H_{0} .\) This proves that more than three-quarters of the school engaged in bullying behavior.

A researcher plans to conduct a significance test at the \(\alpha=0.01\) significance level. She designs her study to have a power of 0.90 at a particular alternative value of the parameter of interest. The probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power is (a) 0.01. (b) 0.10. (c) 0.89. (d) 0.90. (e) 0.99.

Lefties Simon reads a newspaper report claiming that 12% of all adults in the United States are left-handed. He wonders if 12% of the students at his large public high school are left-handed. Simon chooses an SRS of 100 students and records whether each student is right- or left-handed.

The most important condition for sound conclusions from statistical inference is that (a) the data come from a well-designed random sample or randomized experiment. (b) the population distribution be exactly Normal. (c) the data contain no outliers. (d) the sample size be no more than 10% of the population size. (e) the sample size be at least 30.
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