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Chapter 11: Problem 47

Female participation in defense services? When people participating in recent surveys were asked if women should actively participate in defense services, about \(91 \%\) of females and \(91 \%\) of males answered yes and the rest answered no. a. For males and for females, report the conditional distributions on this response variable in a \(2 \times 2\) table, using outcome categories (yes, no). b. If results for the entire population are similar to these, does it seem possible that gender and opinion about having active participation of women in defense services are independent? Explain.

Short Answer

Expert verified
a. 2x2 tables: Females - Yes 91%, No 9%; Males - Yes 91%, No 9%. b. The identical distributions suggest gender and opinion may be independent.

Step by step solution

01

Understand the Problem

We are given survey results indicating that about 91% of both males and females believe women should actively participate in defense services, with the remaining 9% disagreeing. We will create a conditional distribution table for each gender.
02

Create the Table for Conditional Distribution

We will construct a 2x2 table for each gender (male and female) showing the distribution of responses. For females: "Yes" = 91%, "No" = 9%. For males: "Yes" = 91%, "No" = 9%.
03

Constructing the Table

The tables for both males and females will be structured as follows: Female Responses | Yes: 91% | No: 9% Male Responses | Yes: 91% | No: 9%
04

Analyze Independence

Independence implies that the probability distribution across categories is the same regardless of gender. Since both males and females show identical distributions (91% Yes, 9% No), the data suggests gender does not influence opinion on women's participation in defense services.

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Key Concepts

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

Surveys
Surveys play a critical role in gathering information about people's opinions and behaviors. They are data collection tools designed to collect information from a sample of individuals. By using surveys, researchers can get insights into the preferences, beliefs, or experiences of different demographic groups. In the survey mentioned, both males and females were asked their opinions about women participating in defense services. The usage of a survey here is strategic because:
  • It allows for the rapid collection of data from a large number of participants.
  • Surveys can be distributed easily, whether online, face-to-face, or via telephone.
  • The structured format ensures that the same questions are asked to all participants, enhancing the reliability of the results.
By interpreting the results from the survey, researchers can understand the broader public opinion and potentially guide policy-making decisions. In our exercise, both genders showed a strong preference for active female participation in defense services, reflecting a consensus in this sample group.
Gender Independence
Gender independence in statistical terms means that gender does not affect the outcome of the measured variable, which in this case is the opinion on women鈥檚 participation in defense services. When analyzing survey data, we explore whether two variables are independent, meaning one does not influence the other.
  • If gender and opinion are independent, the distribution of opinions ("Yes" or "No") should be identical across gender groups.
  • The concept of independence is key in interpreting the significance of survey results.
In the given exercise, both genders reported 91% in favor of allowing women to participate in defense services with only 9% against this. Since their responses align identically, it suggests that gender does not play a role in shaping these opinions. This can inform us that attitudes towards this topic may be universally shared across genders, at least within the sample surveyed.
Contingency Tables
Contingency tables, also known as cross-tabulations, are used to analyze the relationship between two categorical variables. They provide a clear way to display the frequency distribution of the variables, helping to determine potential independence or correlation. For this exercise:
  • A 2x2 contingency table is constructed for each gender, categorizing responses as "Yes" or "No" to the survey question.
  • Each cell within the table shows the proportion or percentage of the total responses that fall into each category combination.
  • By examining these tables, one can swiftly see that the outcomes are identical for males and females: 91% "Yes" and 9% "No".
The contingency table approach is a powerful visualization and calculation tool in statistics, as it allows easy comparison and assists in analyzing whether there are any observable patterns or relationships in the data. In this scenario, the tables support the finding of gender independence regarding opinions on women's participation in defense services.

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Most popular questions from this chapter

Herbs and the common cold A recent randomized experiment of a multiherbal formula (Immumax) containing echinacea, garlic, ginseng, zinc, and vitamin C was found to improve cold symptoms in adults over a placebo group. "At the end of the study, eight ( \(39 \%\) ) of the placebo recipients and \(18(60 \%)\) of the Immumax recipients reported that the study medication had helped improve their cold symptoms (chi-squared P-value \(=0.01\) )." (M. Yakoot et al., International Joumal of General Medicine, vol. 4,2011 , pp. \(45-51\) ). a. Identify the response variable and the explanatory variable and their categories for the \(2 \times 2\) contingency table that provided this particular analysis.

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.

In the GSS, subjects who were married were asked about the happiness of their marriage, the variable coded as HAPMAR. a. Go to the GSS website sda.berkeley.edu/GSS/, click GSS with no weight as the default, and construct a contingency table for 2012 relating family income (measured as in Table 11.1 ) to marital happiness: Enter FINRELA(r:4;3;2) as the row variable (where \(4=\) "above average," \(3=\) "average, \("\) and \(2=\) "below average") and HAPMAR(r:3;2;1) as the column variable \((3=\) not \(t o 0,2=\) pretty \(,\) and \(1=\) very happy \()\) As the selection filter, enter YEAR(2012). Under Output Options, put a check in the row box (instead of in the column box) for the Percentaging option and put a check in the Summary Statistics box further below. Click on Run the Table. b. Construct a table or graph that shows the conditional distributions of marital happiness, given family income. How would you describe the association? c. Compare the conditional distributions to those in Table \(11.2 .\) For a given family income, what tends to be higher, general happiness or marital happiness for those who are married? Explain.

Serious side effects In \(2007,\) the FDA announced that the popular drug Zelnorm used to treat irritable bowel syndrome was withdrawn from the market, citing concerns about serious side effects. The analysis of several studies revealed that 13 of 11,614 patients receiving the drug had a serious side effect such as heart attack or stroke, compared to just 1 out of 7,031 on placebo. a. Construct a \(2 \times 2\) contingency table from these data. b. Compute the relative risk and interpret. c. Compute the odds ratio and interpret. d. In a letter to health care professionals, the drug maker quoted a P-value of 0.024 for the comparison between the probability of a serious side effect in the drug and placebo group. By entering the data into the Fisher's Exact Test web app (or other software), show that this is the P-value for Fisher's exact test with a two-sided alternative hypothesis. What are the conclusions at a 0.05 significance level?

Variability of chi-squared For the chi-squared distribution, the mean equals \(d\) f and the standard deviation equals \(\sqrt{2(d f)}\) a. Explain why, as a rough approximation, for a large \(d f\) value, \(95 \%\) of the chi-squared distribution falls within \(d f \pm 2 \sqrt{2(d f)}\) b. With \(d f=8,\) show that \(d f \pm 2 \sqrt{2(d f)}\) gives the interval (0,16) for approximately containing \(95 \%\) of the distribution. Using the chi- squared table, show that exactly \(95 \%\) of the distribution actually falls between 0 and 15.5 .
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