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Chapter 11: Problem 67

True or false: \(\boldsymbol{X}^{2}=\mathbf{0}\). The null hypothesis for the test of independence between two categorical variables is \(\mathrm{H}_{0}: X^{2}=0,\) for the sample chi-squared statistic \(X^{2}\). (Hint: Do hypotheses refer to a sample or the population?)

Short Answer

Expert verified
False. The null hypothesis should relate to population parameters, not sample statistics.

Step by step solution

01

Understand the Chi-Squared Test

The chi-squared test is a statistical method used to test the relationship between two categorical variables. It evaluates whether the distribution of sample categorical data matches an expected distribution or tests the independence between variables.
02

Identify Null Hypothesis

The null hypothesis in a chi-squared test for independence typically states that there is no association between the two categorical variables. This is mathematically expressed as the chi-squared statistic being zero, which should symbolically be noted as the population parameter.
03

Population vs. Sample

The hypothesis regards the population parameter rather than sample statistics. Thus, the correct null hypothesis should be about the population chi-squared ( Chi^2), not the sample statistic of chi-squared (X^2).
04

Clarify Difference

Since hypotheses are generally expressed in terms of population parameters (not sample statistics), stating the null hypothesis as "X^2=0" is incorrect. Instead, it should be about the population characteristic, like "Chi^2=0."
05

Conclusion

The statement that "the null hypothesis is X^2=0" conflates the sample statistic with what hypotheses should typically address (the population parameter). Therefore, it is false.

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Key Concepts

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

Chi-Squared Test
The Chi-Squared Test is a statistical tool used to assess the relationship between two categorical variables. This type of test helps determine if observed sample data matches an expected distribution. There are two main varieties of Chi-Squared Tests:
  • Test of Independence: Used to evaluate if two variables are independent or associated.
  • Goodness of Fit Test: Used to see how well sample data fits into a distribution expected by a known population.
During the test, the chi-squared statistic is calculated which assesses the difference between observed and expected frequencies. If these frequencies significantly differ, the null hypothesis is often rejected, suggesting a potential association between the variables.

The simplicity in calculation makes the Chi-Squared Test widely applicable and essential across various research fields.
Null Hypothesis
The Null Hypothesis is a fundamental concept in hypothesis testing, represented as \( H_0 \). For a Chi-Squared Test of independence, the null hypothesis typically posits that there is no relationship between the two variables being examined; they are independent.

For example, if testing whether a new teaching method impacts student grades, the null hypothesis might state that the method has no effect on the grades, suggesting the differences in grades are due to chance.

The process for utilizing the null hypothesis includes:
  • Formulate Null Hypothesis: Specify a statement for no effect or no difference.
  • Collect and Analyze Data: Use statistical tests like the Chi-Squared Test.
  • Decision Making: Compare the calculated statistic with a critical value to decide whether to reject the null hypothesis.
Rejecting the null hypothesis suggests that a variable may influence or be associated with another variable, worthy of further investigation.
Population Parameter
A Population Parameter is a fixed numerical value describing a characteristic of an entire population. Unlike sample statistics that describe only a subset of the population, parameters give a true measure without sampling variability.

In hypothesis testing, the goal is to make inferences about the population parameter using sample data. For instance, in the Chi-Squared Test scenario, we're interested in understanding the population's standard of independence, and we use the sample's chi-squared statistic to infer it.

To better understand this:
  • Population vs. Sample: The entire group being studied versus a smaller group selected from it.
  • Parameter vs. Statistic: A parameter is about the population, and a statistic is about a sample from the population.
The distinction is important since hypotheses should address the population parameters, not just the specifics of the sample data. In Chi-Squared analysis, hypotheses are structured around the population parameter being zero, conveying a perfect independence in the population.

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Most popular questions from this chapter

Happiness and highest degree The table shows 2008 GSS data on happiness and the highest degree attained. a. The chi-squared test of independence has \(X^{2}=64.41\). What conclusion would you make using a significance level of 0.05 ? Interpret. b. Does this large chi-squared value mean there is a strong association between happiness and highest degree? Explain. c. Estimate the difference between the lowest and highest education groups in the proportion who report being not too happy. Interpret. d. Find and interpret the relative risk of being not too happy, comparing the lowest and highest education groups. Interpret.

What is independent of happiness? Which one of the following variables would you think most likely to be independent of happiness: belief in an afterlife, family income, quality of health, region of the country in which you live, satisfaction with job? Explain the basis of your reasoning.

Normal and chi-squared with \(d f=1\) When \(d f=1\), the P-value from the chi- squared test of independence is the same as the P-value for the two-sided test comparing two proportions with the \(z\) test statistic. This is because of a direct connection between the standard normal distribution and the chi-squared distribution with \(d f=1\) : Squaring a \(z\) -score yields a chi-squared value with \(d f=1\) having chi- squared right-tail probability equal to the twotail normal probability for the \(z\) -score. a. Illustrate this with \(z=1.96,\) the \(z\) -score with a twotail probability of \(0.05 .\) Using the chi-squared table or software, show that the square of 1.96 is the chisquared score for \(d f=1\) with a P-value of 0.05 . b. Show the connection between the normal and chisquared values with \(\mathrm{P}\) -value \(=0.01\)

Explaining Fisher's exact test \(\quad\) A pool of six candidates for three managerial positions includes three females and three males. Denote the three females by \(\mathrm{F} 1, \mathrm{~F} 2, \mathrm{~F} 3\) and the three males by \(\mathrm{M} 1, \mathrm{M} 2, \mathrm{M} 3 .\) The result of choosing three individuals for the managerial positions is \((\mathrm{F} 2, \mathrm{M} 1, \mathrm{M} 3)\) a. Identify the 20 possible samples that could have been selected. Explain why the contingency table relating gender to whether chosen for the observed sample is : b. Let \(\hat{p}_{1}\) denote the sample proportion of males selected and \(\hat{p}_{2}\) the sample proportion of females. For the observed table, \(\hat{p}_{1}-\hat{p}_{2}=(2 / 3)-(1 / 3)=1 / 3\). Of the 20 possible samples, show that 10 have \(\hat{p}_{1}-\hat{p}_{2} \geq 1 / 3\). (Note that, if the three managers were randomly selected, the probability would equal \(10 / 20=0.50\) of obtaining \(\hat{p}_{1}-\hat{p}_{2} \geq 1 / 3\). This is the reasoning that provides the one-sided P-value for Fisher's exact test with \(\mathrm{H}_{a}: p_{1}>p_{2} .\)

Life after death and gender In the \(2008 \mathrm{GSS}, 620\) of 809 males and 835 of 978 females indicated a belief in life after death. (Source: Data from CSM, UC Berkeley.) a. Construct a \(2 \times 2\) contingency table relating gender of respondent (SEX, categories male and female) as the rows to belief about life after death (POSTLIFE, categories yes and no) as the columns. b. Find the four expected cell counts for the chisquared test. Compare them to the observed cell counts, identifying cells having more observations than expected. c. The data have \(X^{2}=22.36 .\) Set up its calculation by showing how to substitute the observed and expected cell counts you found into its formula.
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