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Chapter 11: Problem 79

True or false: Statistical but not practical significance Even when the sample conditional distributions in a contingency table are only slightly different, when the sample size is very large it is possible to have a large \(X^{2}\) statistic and a very small P-value for testing \(\mathrm{H}_{0}\) independence.

Short Answer

Expert verified
True. Large sample sizes can make slight differences statistically significant without being practically significant.

Step by step solution

01

Understanding Statistical Significance

Statistical significance refers to the likelihood of an observed effect, such as a difference in conditional distributions, not occurring by random chance. The chi-square (X^2) statistic is used to assess this.
02

Interpreting Practical Significance

Practical significance considers whether the effect size is large enough to have meaningful implications in real-world situations, even if it is statistically significant.
03

Impact of Sample Size on X^2 Test

When the sample size is very large, the X^2 test can yield a large statistic, suggesting statistical significance, even if the differences between groups (conditional distributions) are minute.
04

Evaluating the Given Statement

The statement refers to a scenario where, due to a large sample size, a statistically significant result is obtained that does not have practical significance. This is a common occurrence in hypothesis testing where statistical power is high.

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Key Concepts

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

Practical Significance
While statistical significance tells us if an effect exists, practical significance helps us understand if this effect is meaningful or useful in the real world. Imagine finding a slight difference in average test scores between two study methods. It might be statistically significant, but if it means just a 0.1 point difference, is it worth changing how students are taught? This is where practical significance comes into play. Here are some aspects to consider:
  • **Effect Size:** A key measure in determining practical significance. It helps us measure the strength of an effect or the relationship between variables. Larger effect sizes typically indicate more practical significance.
  • **Real-World Impact:** Even if an effect is statistically significant, we need to ask if it鈥檚 large enough to matter in practice. Does it justify a change in policy or approach?
  • **Context Considerations:** The context in which a study occurs can greatly affect what is practically significant. Parameters for what's considered meaningful can vary across different fields such as medicine, education, or business.
Practical significance provides a broader picture and helps in making informed decisions beyond mere numbers.
Sample Size
Sample size plays a crucial role in statistical analysis and can greatly influence the results of hypothesis tests. It represents the number of observations or data points used in a study. Here's why understanding sample size is important:
  • **Greater Accuracy:** Larger sample sizes generally provide more accurate estimates and increase the reliability of the results. They reduce the margin of error, giving a clearer picture of the population.
  • **Statistical Significance:** In hypothesis testing, larger samples can make it easier to detect statistical significance even for small differences. They tend to produce smaller p-values, which indicates that the results are unlikely due to random chance.
  • **Power of a Test:** A larger sample size increases the test's power, which is the probability of correctly rejecting a false null hypothesis. This means that you are more likely to detect a true effect when it exists.
However, having a very large sample can sometimes lead to finding statistical significance without practical significance, emphasizing the importance of considering both aspects in analysis.
Chi-Square Test
The chi-square test is a statistical method used to determine if there is a significant association between two categorical variables. It鈥檚 often used in contexts like genetics, marketing, or any field with categorical data. Here are key elements of the chi-square test:
  • **Purpose:** It checks if the observed frequencies in a contingency table differ from the frequencies expected under a null hypothesis of no association.
  • **Chi-Square Statistic:** Calculated using the formula \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \(O_i\) are the observed frequencies, and \(E_i\) are the expected frequencies if the variables were independent.
  • **P-Value:** The chi-square statistic helps determine the p-value, indicating the probability of observing an effect given that the null hypothesis is true. A small p-value (typically < 0.05) suggests evidence against the null hypothesis.
  • **Applications:** Widely used in surveys or experiments to test independence or goodness of fit, understanding consumer behavior, or comparing categorical data across different samples.
By considering these aspects, researchers can effectively use the chi-square test to explore relationships between categorical variables and make informed conclusions about their data.

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Most popular questions from this chapter

Babies and gray hair A young child wonders what causes women to have babies. For each woman who lives on her block, she observes whether her hair is gray and whether she has young children, with the results shown in the table that follows. a. Construct the \(2 \times 2\) contingency table that cross-tabulates gray hair (yes, no) with has young children (yes, no) for these nine women. b. Treating has young children as the response variable, obtain the conditional distributions for those women who have gray hair and for those who do not. Does there seem to be an association? c. Noticing this association, the child concludes that not having gray hair is what causes women to have children. Use this example to explain why association does not necessarily imply causation. \begin{tabular}{lcc} \hline Woman & Gray Hair & Young Children \\ \hline Andrea & No & Yes \\ Mary & Yes & No \\ Linda & No & Yes \\ Jane & No & Yes \\ Maureen & Yes & No \\ Judy & Yes & No \\ Margo & No & Yes \\ Carol & Yes & No \\ Donna & No & Yes \\ \hline \end{tabular}

Another predictor of happiness? Go to sda.berkeley.edu/ GSS and find a variable that is associated with happiness, other than variables used in this chapter. Use methods from this chapter to describe and make inferences about the association, in a one-page report.

A study used 1496 patients suffering from low levels of anxiety. The study randomly assigned each subject to a cognitive behavioral therapy (CBT) treatment or a placebo treatment. In this study, increased anxiety levels were observed for 45 of the 729 subjects taking a placebo and for 29 of the 767 subjects taking CBT. a. Report the data in the form of a \(2 \times 2\) contingency table. b. Show how to carry out all five steps of the null hypothesis that having an anxiety attack is not associated with whether one is taking a placebo or CBT. (You should get a chi-squared statistic equal to \(4.5 .\) ) Interpret.

Women's role A recent GSS presented the statement, "Women should take care of running their homes and leave running the country up to men," and \(14.8 \%\) of the male respondents agreed. Of the female respondents, \(15.9 \%\) agreed. Of respondents having less than a high school education, \(39.0 \%\) agreed. Of respondents having at least a high school education, \(11.7 \%\) agreed. a. Report the difference between the proportion of males and the proportion of females who agree. b. Report the difference between the proportion at the low education level and the proportion at the high education level who agree. c. Which variable, gender or educational level, seems to have the stronger association with opinion? Explain your reasoning.

True or false: \(X^{2}=0\) 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?)
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