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Chapter 4: Problem 42

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. \(H_{0}: \rho=0 \quad\) vs \(H_{a}: \rho \neq 0\)

Short Answer

Expert verified
The sample statistic we might record for each simulated sample to create the randomization distribution in this scenario is the sample correlation coefficient, \( r \).

Step by step solution

01

Understanding the Hypotheses

Here, we have two hypotheses about the population correlation coefficient \( \rho \). The null hypothesis \( H_0 \) is that \( \rho = 0 \), which means there is no correlation between the two variables. The alternative hypothesis \( H_a \) is that \( \rho \neq 0 \), indicating there is a correlation, either positive or negative.
02

Decide on the Sample Statistic

For the sample statistic, we need something that can help us estimate the correlation in our sample data and compare it to the hypothesized value. The appropriate statistic in this case is the sample correlation coefficient, \( r \). This statistic can provide a measure of the strength and direction of the relationship between the two variables in the sample data.
03

Creating the Randomization Distribution

We would use this \( r \) value as our test statistic and record its value for each simulated sample. By doing this, we can create a randomization distribution of \( r \) values under the assumption that the null hypothesis is true (i.e., that there is no correlation). This distribution can then be used to determine the probability of observing a result as extreme as our sample data if the null hypothesis is true, which forms the basis for deciding whether to reject the null hypothesis.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis is the statement we initially assume to be true. It acts as a starting point for our investigation. When dealing with correlation, the null hypothesis, denoted as \( H_0 \), often posits no effect or no relationship.
In the provided example, the null hypothesis \( H_0 \) is \( \rho = 0 \). This indicates that the true population correlation coefficient is zero, meaning there's no association between the two variables being studied.
The essence of the null hypothesis is to offer a statement for testing. By assuming there's no effect or difference, researchers have a benchmark to measure any observed effect against.
Alternative Hypothesis
The alternative hypothesis, represented as \( H_a \), is a statement that directly contradicts the null hypothesis. It suggests that there is a real effect or relationship present in the data. This hypothesis serves as the rival to the null hypothesis, proposing an opposing view that researchers seek to provide evidence for.
In our correlation example, the alternative hypothesis \( H_a \) is \( \rho eq 0 \). This claim suggests that a correlation does exist, be it positive or negative. Here, the goal is to collect enough evidence from the data to support the alternative hypothesis over the null hypothesis.
  • Positive correlation: If \( \rho > 0 \), it implies that as one variable increases, so does the other.
  • Negative correlation: If \( \rho < 0 \), it means one variable decreases as the other increases.
Through testing, if the data shows significant evidence against the null hypothesis, the alternative hypothesis gains support.
Sample Statistic
A sample statistic is a numerical value calculated from sample data. It is used to estimate and infer population parameters. In hypothesis testing, the sample statistic provides insight into the population aspect we are examining.
In the context of testing for correlation, the sample correlation coefficient, \( r \), is used as the sample statistic. This coefficient measures the strength and direction of the linear relationship between two sample variables. The calculation of \( r \) helps assess how close the sample data aligns with the null hypothesis.
Once \( r \) is computed from a sample, it can be used to create a randomization distribution. This distribution helps determine the likelihood of observing the sample correlation under the null hypothesis. If the observed \( r \) significantly deviates from what would be expected under \( H_0 \), it may indicate that the null hypothesis is not accurate, giving weight to the alternative hypothesis.

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Most popular questions from this chapter

Scientists studying lion attacks on humans in Tanzania \(^{32}\) 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.)

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: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=42 / 100=0.42\) with \(n=100\)

Influencing Voters Exercise 4.39 on page 272 describes a possible study to see if there is evidence that a recorded phone call is more effective than a mailed flyer in getting voters to support a certain candidate. The study assumes a significance level of \(\alpha=0.05\) (a) What is the conclusion in the context of thisstudy if the p-value for the test is \(0.027 ?\) (b) In the conclusion in part (a), which type of error are we possibly making: Type I or Type II? Describe what that type of error means in this situation. (c) What is the conclusion if the p-value for the test is \(0.18 ?\)

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing to see if a well-known company is lying in its advertising. If there is evidence that the company is lying, the Federal Trade Commission will file a lawsuit against them.

A study \(^{20}\) conducted in June 2015 examines ownership of tablet computers by US adults. A random sample of 959 people were surveyed, and we are told that 197 of the 455 men own a tablet and 235 of the 504 women own a tablet. We want to test whether the survey results provide evidence of a difference in the proportion owning a tablet between men and women. (a) State the null and alternative hypotheses, and define the parameters. (b) Give the notation and value of the sample statistic. In the sample, which group has higher tablet ownership: men or women? (c) Use StatKey or other technology to find the pvalue.
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