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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

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.6\) vs \(H_{a}: p>0.6\) Sample data: \(\hat{p}=52 / 80=0.65\) with \(n=80\)

In Exercises 4.107 to \(4.111,\) null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. $$ H_{0}: \rho=0 \text { vs } H_{a}: \rho \neq 0 $$

Pesticides and ADHD Are children with higher exposure to pesticides more likely to develop ADHD (attention-deficit/hyperactivity disorder)? In a recent study, authors measured levels of urinary dialkyl phosphate (DAP, a common pesticide) concentrations and ascertained ADHD diagnostic status (Yes/No) for 1139 children who were representative of the general US population. \(^{8}\) The subjects were divided into two groups based on high or low pesticide concentrations, and we compare the proportion with ADHD in each group. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) In the sample, children with high pesticide levels were more likely to be diagnosed with ADHD. Can we necessarily conclude that, in the population, children with high pesticide levels are more likely to be diagnosed with ADHD? (Whether or not we can make this generalization is, in fact, the statistical question of interest.) (c) To assess statistical significance, we assume the null hypothesis is true. What does that mean in this case? State your answer in terms of pesticides and ADHD. (d) The study found the results to be statistically significant. Which of the hypotheses, \(H_{0}\) or \(H_{a}\), is no longer a very plausible possibility? (e) What do the statistically significant results imply about pesticide exposure and ADHD?

Rolling Dice You roll a die 60 times and record the sample proportion of fives, and you want to test whether the die is biased to give more fives than a fair die would ordinarily give. To find the p-value for your sample data, you create a randomization distribution of proportions of fives in many simulated samples of size 60 with a fair die. (a) State the null and alternative hypotheses. (b) Where will the center of the distribution be? Why? (c) Give an example of a sample proportion for which the number of 5 's obtained is less than what you would expect in a fair die. (d) Will your answer to part (c) lie on the left or the right of the center of the randomization distribution? (e) To find the p-value for your answer to part (c), would you look at the left, right, or both tails? (f) For your answer in part (c), can you say anything about the size of the p-value?

Do iPads Help Kindergartners Learn: A Series of Tests Exercise 4.163 introduces a study in which half of the kindergarten classes in a school district are randomly assigned to receive iPads. We learn that the results are significant at the \(5 \%\) level (the mean for the iPad group is significantly higher than for the control group) for the results on the HRSIW subtest. In fact, the HRSIW subtest was one of 10 subtests and the results were not significant for the other 9 tests. Explain, using the problem of multiple tests, why we might want to hesitate before we run out to buy iPads for all kindergartners based on the results of this study.
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