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Chapter 4: Problem 71

In Exercises 4.71 to \(4.74,\) using the p-value given, are the results significant at a \(10 \%\) level? At a \(5 \%\) level? At a 1\% level? $$ \text { p-value }=0.0320 $$

Short Answer

Expert verified
The results are significant at the 10% and 5% levels, but not at the 1% level.

Step by step solution

01

Comparison with 10% level

The 10% significance level means that there is a 10% (or 0.10) chance that you would find a relationship between variables when there is no relationship. Compare 0.10 with the p-value 0.0320. The p-value 0.0320 is less than the 10% level (0.10), so the results are significant at a 10% level.
02

Comparison with 5% level

Now, we compare the p-value with the 5% level, which means that there is a 5% (or 0.05) chance that you would find a relationship between variables when there is no relationship. Comparing 0.05 with our p-value 0.0320 shows that our p-value is less than the 5% level (0.05). So, the results are also significant at the 5% level.
03

Comparison with 1% level

Let's compare the p-value with the 1% level, which means that there is a 1% (or 0.01) chance that you would find a relationship between variables when there is no relationship. The p-value 0.0320 is greater than the 1% level (0.01). Therefore, the results are not significant at the 1% level.

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Key Concepts

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

Understanding the p-value
The p-value is a crucial concept in statistics when it comes to determining the strength of your findings in a hypothesis test.
It helps us decide whether to reject the null hypothesis. In simple terms, the p-value is the probability that the observed data would occur if the null hypothesis were true.
  • It's a measure of evidence against the null hypothesis.
  • A lower p-value indicates stronger evidence against the null hypothesis.
  • If the p-value is very low, it suggests the results are unlikely under the null hypothesis, pointing toward rejecting it.
In this exercise, a p-value of 0.0320 was provided. This means there's a 3.2% chance the observed data could occur if the null hypothesis is true.
This percentage is used to measure the strength of our evidence.
What is a Significance Level?
The significance level, often denoted as \( \alpha \), is the threshold we set for deciding whether to reject the null hypothesis.
It's the probability of rejecting the null hypothesis when it is actually true, known as a Type I error.
  • Common significance levels are 10% (0.10), 5% (0.05), and 1% (0.01).
  • A lower significance level means stricter criteria for rejecting the null hypothesis.
  • Choosing the right level depends on the context of your test and the consequences of errors.
In the problem, we compared the p-value to different significance levels:
  • At the 10% level, 0.0320 < 0.10, so the result is significant.
  • At the 5% level, 0.0320 < 0.05, confirming significance.
  • At the 1% level, 0.0320 > 0.01, indicating insignificance.
This comparison shows how significance levels guide our inference options.
The Process of Hypothesis Testing
Hypothesis testing is a method used to determine if a certain claim about a parameter is supported by data.
We start by stating the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \).
  • The null hypothesis typically represents no effect or no difference.
  • The alternative hypothesis represents the presence of an effect or difference.
The testing involves calculating the p-value and comparing it to the chosen significance level.
Here鈥檚 a basic outline:
  • Step 1: Determine null \( H_0 \) and alternative \( H_a \) hypotheses.
  • Step 2: Choose a significance level \( \alpha \).
  • Step 3: Collect and analyze data to calculate the p-value.
  • Step 4: Compare p-value with \( \alpha \).
  • Step 5: Decide whether to reject or fail to reject \( H_0 \).
In our example, the hypothesis test demonstrated that the p-value leads to rejecting \( H_0 \) for both 10% and 5% significance levels, but not for 1%, highlighting its practical application.

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Most popular questions from this chapter

Red Wine and Weight Loss Resveratrol, an ingredient in red wine and grapes, has been shown to promote weight loss in rodents. A recent study \(^{19}\) investigates whether the same phenomenon holds true in primates. The grey mouse lemur, a primate, demonstrates seasonal spontaneous obesity in preparation for winter, doubling its body mass. A sample of six lemurs had their resting metabolic rate, body mass gain, food intake, and locomotor activity measured for one week prior to resveratrol supplementation (to serve as a baseline) and then the four indicators were measured again after treatment with a resveratrol supplement for four weeks. Some p-values for tests comparing the mean differences in these variables (before vs after treatment) are given below. In parts (a) to (d), state the conclusion of the test using a \(5 \%\) significance level, and interpret the conclusion in context. (a) In a test to see if mean resting metabolic rate is higher after treatment, \(p=0.013\). (b) In a test to see if mean body mass gain is lower after treatment, \(p=0.007\) (c) In a test to see if mean food intake is affected by the treatment, \(p=0.035\). (d) In a test to see if mean locomotor activity is affected by the treatment, \(p=0.980\) (e) In which test is the strongest evidence found? The weakest? (f) How do your answers to parts (a) to (d) change if the researchers make their conclusions using a stricter \(1 \%\) significance level? (g) For each p-value, give an informal conclusion in the context of the problem describing the level of evidence for the result. (h) The sample only included six lemurs. Do you think that we can generalize to the population of all lemurs that body mass gain is lower on average after four weeks of a resveratrol supplement? Why or why not?

Indicate whether the analysis involves a statistical test. If it does involve a statistical test, state the population parameter(s) of interest and the null and alternative hypotheses. Polling 1000 people in a large community to determine if there is evidence for the claim that the percentage of people in the community living in a mobile home is greater than \(10 \%\)

Weight Loss Program Suppose that a weight loss company advertises that people using its program lose an average of 8 pounds the first month and that the Federal Trade Commission (the main government agency responsible for truth in advertising) is gathering evidence to see if this advertising claim is accurate. If the FTC finds evidence that the average is less than 8 pounds, the agency will file a lawsuit against the company for false advertising. (a) What are the null and alternative hypotheses the FTC should use? (b) Suppose that the FTC gathers information from a very large random sample of patrons and finds that the average weight loss during the first month in the program is \(\bar{x}=7.9\) pounds with a p-value for this result of \(0.006 .\) What is the conclusion of the test? Are the results statistically significant? (c) Do you think the results of the test are practically significant? In other words, do you think patrons of the weight loss program will care that the average is 7.9 pounds lost rather than 8.0 pounds lost? Discuss the difference between practical significance and statistical significance in this context.

Penalty Shots in Soccer A recent article noted that it may be possible to accurately predict which way a penalty-shot kicker in soccer will direct his shot. \({ }^{23}\) The study finds that certain types of body language by a soccer player-called "tells"-can be accurately read to predict whether the ball will go left or right. For a given body movement leading up to the kick, the question is whether there is strong evidence that the proportion of kicks that go right is significantly different from one-half. (a) What are the null and alternative hypotheses in this situation? (b) If sample results for one type of body movement give a p-value of \(0.3184,\) what is the conclusion of the test? Should a goalie learn to distinguish this movement? (c) If sample results for a different type of body movement give a p-value of \(0.0006,\) what is the conclusion of the test? Should a goalie learn to distinguish this movement?

Beer and Mosquitoes Does consuming beer attract mosquitoes? Exercise 4.17 on page 232 discusses an experiment done in Africa testing possible ways to reduce the spread of malaria by mosquitoes. In the experiment, 43 volunteers were randomly assigned to consume either a liter of beer or a liter of water, and the attractiveness to mosquitoes of each volunteer was measured. The experiment was designed to test whether beer consumption increases mosquito attraction. The report \(^{27}\) states that "Beer consumption, as opposed to water consumption, significantly increased the activation... of An. gambiae [mosquitoes]... \((P<0.001)\) (a) Is this convincing evidence that consuming beer is associated with higher mosquito attraction? Why or why not? (b) How strong is the evidence for the result? Explain. (c) Based on these results, is it reasonable to conclude that consuming beer causes an increase in mosquito attraction? Why or why not?
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