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Chapter 10: Problem 15

Oil Leaders The table shows the world's five largest crude oil producing countries and their total oil output in percentage for the years 2011 and 2012 (www.whichcountry.co). Give two reasons why you should not do a chi-square test with these data.

Short Answer

Expert verified
The Chi-Square test is not suitable for these data for two reasons. 1. The data from the table is numerical, not categorical. Chi-Square test requires categorical data 2. The observations are not independent, a prerequisite for deploying a Chi-Square Test.

Step by step solution

01

Understand the Chi-Square Test Conditions

Firstly, it should be understood what a chi-square test is and when it should be used. The chi-square test is primarily used to determine if there is a significant association between two categorical variables in a sample. It requires raw data to be placed into a contingency table, where the rows represent one categorical variable and the columns represent another.
02

Reason 1: Data Categories and Chi-Square

One of the reasons not to use the Chi-Square test for the provided data is because it's not categorical but rather numerical. The data given are percentages of total oil output by country for 2011 and 2012. One important prerequisite for a Chi-Square test is that the data must be categorical (nominal or ordinal) but not anywhere interval or ratio. Here, the oil output percentages are interval data because there's an order and exact differences between the measures.
03

Reason 2: Independent Observations Requirement

Another crucial reason not to use Chi-Square test is about the condition of independent observations, which means the value of one observation does not affect/influence the value of another observation. However, the percentage of global oil production by one country could affect the percentage of another country's output; they are not independent. For example, if country A increases its production, the percentage of the world's production attributable to other countries would decrease even if actual production levels remained the same. Hence, the observations are not independent here, which violates the requirement for the Chi-Square test.

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Key Concepts

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

Categorical Data
Categorical data refers to variables that can be divided into groups or categories. These categories are distinct and non-overlapping, such as gender, types of animals, or yes/no responses.
In the case of a Chi-Square Test, it is essential that the data be categorical. This means that each piece of data falls into one category or another without ambiguity about where it belongs.
For instance, if you are testing whether a die is fair, your categories would be the six faces of the die. You would count how many times each face appears and then perform the Chi-Square Test to see if each face appeared an equal number of times.

In the exercise provided, the data are numerical percentages, not categorical data, which makes them unsuitable for a Chi-Square Test.
This is because percentages are continuous data types that can take any value within a range, rather than representing discrete categories.
Independent Observations
The concept of independent observations is crucial in statistical analysis and specifically for executing a Chi-Square Test.
Independent observations mean that the occurrence of one event does not affect the occurrence of another. This assumption ensures that there is no influence or dependency between the data points.
It helps to ensure the validity and reliability of your statistical test.
For example, in a medical study evaluating the effectiveness of a drug, each participant receiving the treatment should do so independently of all others. Their response to the medication must not affect or be affected by others' responses.

However, in the oil data example, if one country's output changes, it directly influences the percentage share of the other countries, violating the independence condition.
This interlinking makes the data non-independent and thus inappropriate for a Chi-Square Test.
Statistical Analysis
Statistical analysis involves collecting, exploring, and interpreting large amounts of data to uncover patterns and trends.
It employs various techniques and tests to analyze specific types of data, with adherence to underlying assumptions crucial for valid results.
When it comes to using statistical analysis techniques like the Chi-Square Test, certain conditions must be met, such as the presence of categorical data and independent observations.

For effective statistical analysis, it's important to choose the right test for your data type.
  • If your data is categorical, check the independence of observations, and use Chi-Square.
  • For numerical data, consider other tests like t-tests or ANOVA.
Understanding your data's nature and the conditions of the tests allows you to derive meaningful and accurate insights efficiently.

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Most popular questions from this chapter

Drug for Asthma (Example 8) Eosinophils are a form of white blood cell that is often present in people suffering from allergies. People with asthma and high levels of eosinophils who used steroid inhalers were given either a new drug or a placebo. Read extracts from the abstract of this study that appear below, and then evaluate the study. See page 539 for questions and guidance. "Methods: We enrolled patients with persistent, moderate-tosevere asthma and a blood eosinophil count of at least 300 cells per microliter ... who used medium-dose to high-dose inhaled glucocorticoids.... We administered dupilumab \((300 \mathrm{mg})\) or placebo subcutaneously once weekly. The primary end point was the occurrence of an asthma exacerbation [worsening]. Results: A total of 52 patients were [randomly] assigned to the dupilumab group, and 52 patients were [randomly] assigned to the placebo group..... Three patients had an asthma exacerbation with dupilumab \((6 \%)\) versus 23 with placebo \((44 \%)\), corresponding to an \(87 \%\) reduction with dupilumab (odds ratio, \(0.08 ; 95 \%\) confidence interval, \(0.02\) to \(0.28 ; \mathrm{P}<0.001)\). Conclusions: In patients with persistent, moderate-to-severe asthma and elevated eosinophil levels who used inhaled glucocorticoids and LABAs, dupilumab therapy, as compared with placebo, was associated with fewer asthma exacerbations [worsenings]."

Literacy Rates World literacy rates for individuals of 15 years of age or older are given in the data table as a percentage. Give two reasons why a chi- square test is not appropriate for this set of data.

One treatment for multiple myeloma (cancer of the blood and bones) is a stem cell transplant. However, in some cases the cancer returns. McCarthy and colleagues reported on a study that randomly assigned 460 patients ( 100 days after a stem cell transplant) to receive either lenalidomide or placebo. At one point in the study, 46 of the patients who received the real drug had a bad result (had progressive disease or had died), compared to 101 of those who received the placebo. Assume that exactly half were assigned to each group. a. Find and compare the percentages that had a bad result for the two groups. b. Test the hypothesis that the drug reduced the chance of a bad result compared to the placebo using a significance level of \(0.05\). c. The study started in April 2005 and was "unblinded" in 2009 when an interim analysis showed better results with the group taking the drug. After the unblinding, many of the patients from the placebo group "crossed over" to the drug group. Explain what you think "unblinding" means and why this seems like a reasonable thing to do. (Source: P. L. McCarthy et al. 2012. Lenalidomide after stem-cell transplantation for multiple myeloma. New England Journal of Medicine 366, 1770-1781.)

Tests a. In Chapter 8 , you learned some tests of proportions. Are tests of proportions used for categorical or numerical data? b. In this chapter, you are learning to use chi-square tests. Do these tests apply to categorical or numerical data?

Odd-Even Formula A survey was taken of a random sample of people noting their gender and asking whether they agreed with the Odd-Even Formula (OEF) to control the alarming levels of air pollution. Minitab results are shown. \(=1\) 2 Chi-square Test for Association: Opinion, Gender ciat:1 Rows: Ooinion Columns: Gender $$ \begin{array}{rr}\text { Male Female } \\ \text { Disagree } & 42 & 44 \\\ 42.17 & 43.83 \\ \text { Agree } & 11.4 & 86 \\ & 110 & 14.17 & \\ & 109.83 & \\ \mathrm{All} & & \\ & 152 & 158 & 310 \\ \text { Cel1 Contents: } & \text { Count } \\ & \text { Expected count } \\ \text { Pearson Chi-Square }=0.002, \mathrm{DF}=1, \text { p-value }=0.966\end{array} $$ a. Find the percentage of men and women in the sample who agreed with the OEF method, and compare these percentages. b. Test the hypothesis that opinions about OEF and gender are independent using a significance level of \(0.05\). c. Does this suggest that men and women have significantly different views about the OEF method?
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