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Chapter 4: Problem 109

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. $$ H_{0}: \rho=0 \text { vs } H_{a}: \rho \neq 0 $$

Short Answer

Expert verified
The appropriate notation for a sample statistic we might record in this case is \(r\), the sample correlation coefficient.

Step by step solution

01

Understanding the Hypotheses

Firstly, the null hypothesis \(H_{0}: \rho=0\) means that there is no correlation between the variables under study in the population. The alternative hypothesis \(H_{a}: \rho \neq 0\) means that there exists a correlation in the population, but it could be either negative or positive.
02

Choosing the Sample Statistic

Considering the nature of the hypotheses, a suitable statistic for the randomization distribution is the sample correlation coefficient, typically denoted as \(r\). The sample correlation coefficient \(r\) quantifies the degree of correlation among data points in a sample and is an appropriate statistic for testing hypotheses about the population correlation \(\rho\).

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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, often represented by the notation \(H_0\), serves as the foundational statement in hypothesis testing. It proposes that there is no significant effect or relationship present in the population being studied. In simpler terms, it suggests that any observations in the data are purely due to chance. For example, in the context of correlation, the null hypothesis \(H_0: \rho = 0\) asserts that there is no correlation between two variables within the population. This hypothesis establishes a baseline so that we can measure whether the data provides enough evidence to suggest otherwise. Hypothesis tests aim to evaluate this baseline, only rejecting it if there is substantial evidence against it.
Alternative Hypothesis
The alternative hypothesis, denoted as \(H_a\), is the statement that we consider if the null hypothesis is rejected. It represents a possible effect or relationship that exists within the population. In the exercise's example, the alternative hypothesis \(H_a: \rho eq 0\) suggests that a correlation exists between the two variables, which could be either positive or negative. This hypothesis is essentially what researchers often set out to demonstrate. If statistical testing provides sufficient evidence, we reject the null in favor of the alternative, indicating that the data supports the presence of an effect or relationship.
Sample Statistic
A sample statistic is a numerical measure calculated from a data sample and is used to estimate the population parameter. In hypothesis testing, particularly for the given exercise, a sample statistic is essential for evaluating the hypotheses. For example, the sample correlation coefficient \(r\) serves as the sample statistic here. It estimates the population correlation coefficient \(\rho\).
  • The sample statistic helps create a randomization distribution.
  • It allows researchers to assess whether observed results are due to random chance or signify a real effect in the population.
  • By comparing \(r\) calculated from your sample data to a distribution generated from simulations, you can determine how probable it is to see such a correlation by chance if the null hypothesis were true.
Thus, the sample statistic is vital in determining whether to reject or accept the null hypothesis based on evidence from the data.

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Most popular questions from this chapter

In Exercises 4.112 to \(4.116,\) the null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p<0.5\) Sample: \(\hat{p}=0.4, n=30\) Randomization statistic \(=\hat{p}\)

Watch Out for Lions after a Full Moon Scientists studying lion attacks on humans in Tanzania \(^{34}\) found that 95 lion attacks happened between \(6 \mathrm{pm}\) and \(10 \mathrm{pm}\) within either five days before a full moon or five days after a full moon. Of these, 71 happened during the five days after the full moon while the other 24 happened during the five days before the full moon. Does this sample of lion attacks provide evidence that attacks are more likely after a full moon? In other words, is there evidence that attacks are not equally split between the two five-day periods? Use StatKey or other technology to find the p-value, and be sure to show all details of the test. (Note that this is a test for a single proportion since the data come from one sample.)

Translating Information to Other Significance Levels Suppose in a two-tailed test of \(H_{0}: \rho=0 \mathrm{vs}\) \(H_{a}: \rho \neq 0,\) we reject \(H_{0}\) when using a \(5 \%\) significance level. Which of the conclusions below (if any) would also definitely be valid for the same data? Explain your reasoning in each case. (a) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(1 \%\) significance level. (b) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(10 \%\) significance level. (c) Reject \(H_{0}: \rho=0\) in favor of the one-tail alternative, \(H_{a}: \rho>0,\) at a \(5 \%\) significance level, assuming the sample correlation is positive.

Eating Breakfast Cereal and Conceiving Boys Newscientist.com ran the headline "Breakfast Cereals Boost Chances of Conceiving Boys," based on an article which found that women who eat breakfast cereal before becoming pregnant are significantly more likely to conceive boys. \({ }^{42}\) The study used a significance level of \(\alpha=0.01\). The researchers kept track of 133 foods and, for each food, tested whether there was a difference in the proportion conceiving boys between women who ate the food and women who didn't. Of all the foods, only breakfast cereal showed a significant difference. (a) If none of the 133 foods actually have an effect on the gender of a conceived child, how many (if any) of the individual tests would you expect to show a significant result just by random chance? Explain. (Hint: Pay attention to the significance level.) (b) Do you think the researchers made a Type I error? Why or why not? (c) Even if you could somehow ascertain that the researchers did not make a Type I error, that is, women who eat breakfast cereals are actually more likely to give birth to boys, should you believe the headline "Breakfast Cereals Boost Chances of Conceiving Boys"? Why or why not?

Definition of a P-value Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.
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