/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Give the rejection region for a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 7

Give the rejection region for a chi-square test of independence if the contingency table involves \(r\) rows and \(c\) columns. $$r=3, c=3, \alpha=.10$$

Short Answer

Expert verified
Answer: The rejection region for a chi-square test of independence in a 3x3 contingency table with a significance level of 0.10 is 饾湌虏 鈮� 7.78.

Step by step solution

01

Calculate degrees of freedom

To find the degrees of freedom (df) for a chi-square test of independence, you need to use the formula: \(df=(r-1)(c-1)\), where \(r\) is the number of rows, and \(c\) is the number of columns. In this case, \(r=3\) and \(c=3\), so the calculation will be: $$ df = (3-1)(3-1) = 2(2) = 4 $$ Hence, the degrees of freedom for this contingency table is 4.
02

Find the critical chi-square value

To determine the critical chi-square (\(\chi_{\alpha}^2\)) value, use the given significance level (\(\alpha=.10\)) and the degrees of freedom calculated in step 1. As the critical chi-square values have to be obtained from a \(\chi^2-\) table, look up the value for \(df=4\) and \(\alpha=.10\) in a chi-square table, which yields a \(\chi_{\alpha}^2\) value of 7.78.
03

Illustrate the rejection region

The rejection region for our chi-square test of independence is the chi-square values that fall on or above the critical chi-square value. Since the critical chi-square value is 7.78, the rejection region for this test with \(\alpha=.10\) is: $$ \chi^2 \ge 7.78 $$ Any chi-square test statistic that results in an obtained value of 7.78 or greater will fall in the rejection region, causing to reject the null hypothesis and conclude that there is a significant relationship between the variables in the contingency table.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Contingency Table
In statistics, a contingency table is a type of table in a matrix format that displays the frequency distribution of the variables. It helps to understand the relationship between two categorical variables.

For example, consider a study examining the relationship between exercise frequency and illness during flu season. One variable would be the frequency of exercise (e.g., 'none', 'occasional', 'regular'), and the other might be incidence of illness ('sick', 'not sick'). A contingency table would show how many individuals fall into each category so that patterns can be detected and analyzed statistically.
Degrees of Freedom
The concept of degrees of freedom (df) in a chi-square test reflects the number of values in the calculation of a statistic that are free to vary. It's a crucial component as it affects the shape and scale of the chi-square distribution used to determine the critical value.

To compute it for a chi-square test of independence, use the formula: \(df = (r-1)(c-1)\), where \(r\) is the number of rows and \(c\) is the number of columns in the contingency table. In our case with a 3x3 table, the degrees of freedom would be 4. This valuable information will later define the threshold of the chi-square statistic for deciding whether to reject the null hypothesis.
Rejection Region
The rejection region in hypothesis testing is the range of values for the test statistic that leads to a decision to reject the null hypothesis. For the chi-square test, this involves comparing the test statistic to a critical value based on the chi-square distribution.

If the calculated chi-square statistic falls into the rejection region, it suggests a low probability that the observed distribution is due to chance, and we therefore reject the null hypothesis. This decision has real-world implications, as it might influence the conclusions we draw from our data about the independence, or association, of our variables.
Critical Chi-Square Value
The critical chi-square value is a threshold determined by the significance level and the degrees of freedom. This value is obtained from the chi-square distribution table.

Using our example with 4 degrees of freedom and a significance level of 0.10, the critical chi-square value is 7.78. This value separates the rejection region (where we would reject the null hypothesis) from the region of acceptance (where we retain the null hypothesis). It's like a cutoff score that determines whether the evidence against the null hypothesis is strong enough to declare a statistically significant finding.
Significance Level
The significance level, denoted by \(\)alpha\), is the threshold for determining whether the null hypothesis can be rejected. It represents the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error.

Common significance levels include 0.05 (5%), 0.01 (1%), and 0.10 (10%). A lower significance level means that you require stronger evidence to reject the null hypothesis. In the given exercise, a significance level of 0.10 implies that we are willing to accept a 10% chance of incorrectly rejecting the null hypothesis of no association between the variables being tested.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Give the rejection region for a chi-square test of independence if the contingency table involves \(r\) rows and \(c\) columns. $$r=2, c=4, \alpha=.05$$

Find the appropriate degrees of freedom for the chisquare test of independence. $$\text { three rows and three columns }$$

Find the appropriate degrees of freedom for the chisquare test of independence. $$\text { five rows and four columns }$$

Manganese nodules are mineral-rich concoctions found abundantly on the deep- sea floor. \({ }^{10}\) A research report relates the magnetic age of the earth's crust to the "probability of finding manganese nodules," giving the number of samples of the earth's core and the percentage of those that contain manganese nodules for each of a set of magnetic-crust ages. Do the data provide sufficient evidence to indicate that the probability of finding manganese nodules in the deepsea earth's crust is dependent on the magnetic- age classification? $$ \begin{array}{lcc} \hline \text { Age } & \begin{array}{l} \text { Number of } \\ \text { Samples } \end{array} & \begin{array}{l} \text { Percentage with } \\ \text { Nodules } \end{array} \\ \hline \text { Miocene-recent } & 389 & 5.9 \\ \text { Oligocene } & 140 & 17.9 \\ \text { Eocene } & 214 & 16.4 \\ \text { Paleocene } & 84 & 21.4 \\ \text { Late Cretaceous } & 247 & 21.1 \\ \text { Early and Middle Cretaceous } & 1120 & 14.2 \\ \text { Jurassic } & 99 & 11.0 \\ \hline \end{array} $$

The "sandwich generation" refers to middle-aged Americans who are either providing support for an aging parent while raising child under 18 or supporting a child over 18 . The information that follows summarizes the results of a poll \(^{8}\) regarding support for a parent 65 years or older by demographic group. $$ \begin{array}{lrrr} \hline \text { Provide Financial } & & & \\ \text { Support } & \text { Yes } & \text { No } & \text { Total } \\ \hline \text { Hispanics } & 134 & 66 & 200 \\ \text { Blacks } & 86 & 114 & 200 \\ \text { Whites } & 49 & 151 & 200 \\ \hline \end{array} $$ a. Find the proportion of individuals providing financial support for their parents for each demographic group. b. Are there significant differences among the proportions providing financial support for their parents for these demographic groups of Americans? Use \(\alpha=.01\). Are your results consistent with the observed proportions in part a?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.