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Chapter 22: Problem 32

Does your doctor know? (part 5) In Exercises 24 ?, 26 ?, 28 ?, and 30 ?, we considered data on articles in the NEJM. The original study listed 23 different statistics methods. (The list read: \(t\) -tests, contingency tables, linear regression, \(\ldots\).) Why would it not be appropriate to use a chi- square test on the \(23 \times 3\) table with a row for each method?

Short Answer

Expert verified
A chi-square test wouldn't be appropriate in this scenario because the data in the table represent different statistical methods, not categories or frequencies of observation. Additionally, without knowing what the three columns represent, one can't establish expected frequencies, which are requisite for a chi-square test.

Step by step solution

01

Understanding Chi-Square Test

Chi-square test is used for comparison between observed and expected frequencies in one or more categories. It is frequently used in hypothesis testing.
02

Identifying the Table Structure

The given table in the exercise is a 23x3 table representing 23 different statistical methods. However, without knowing what exactly the 3 columns represent, it's difficult to specifically determine if a chi-square test is appropriate or not.
03

Identifying the nature of the data

The crucial point to understand is the nature of the data. The 23 different statistical methods are techniques, not categories or frequencies of observation. Therefore, applying a chi-square test, primarily used for categorical data, on techniques used for statistical analysis might not be appropriate.
04

Lack of categorical variables

The exercise doesn't seem to offer categorical variables needed to conduct a chi-square test. It only lists statistical methods. The chi-square test requires observed and expected data, thus without an understanding of what the three columns represent, a chi-square test wouldn't be suitable.

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Key Concepts

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

Statistical Methods
When exploring the world of data analysis, we encounter various Statistical Methods which serve as tools to help us understand and interpret data.

Statistical methods include a range of techniques from the simple to the complex. For instance, a t-test helps us compare the means of two groups, while linear regression can model relationships between variables. They form the backbone of data analysis and are selected based on the type of data and the specific questions researchers aim to answer.

Each statistical method comes with its assumptions and appropriate use cases. So, for example, choosing to apply a method designed for continuous data, such as ANOVA, to categorical data would be misuse of that technique. Similarly, a chi-square test鈥攚hich is inherently designed for categorical data鈥攎ight not be suitable for a situation where statistical methods or techniques themselves are the subjects of analysis.
Hypothesis Testing
The core of many statistical analyses lies in Hypothesis Testing, a systematic process used to evaluate assumptions about a dataset.

The process starts with the formulation of two contradictory hypotheses: the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \). The null hypothesis often suggests that there is no effect or difference, while the alternative hypothesis indicates the presence of an effect or difference.

The chi-square test is a commonly used tool in hypothesis testing when dealing with categorical data. It measures how well observed frequency data match expected frequencies derived from a particular hypothesis. If the result of the chi-square test is significant, it suggests that the observed data do not fit the expected pattern under the null hypothesis, potentially leading to the null hypothesis being rejected in favor of the alternative.
Categorical Data Analysis
In Categorical Data Analysis, we focus on data that can be sorted into categories, such as 'Yes' or 'No' responses, types of plants, or likert scale survey responses. Unlike numerical data, categorical data summarize attributes rather than measure them.

For the analysis of such data, the chi-square test plays a pivotal role by comparing observed counts in categorical variables to expected counts if there were no association between the variables. However, it is crucial to only apply this test when the data is truly categorical, and there are sufficient observations to meet the test's requirements.

An inappropriately chosen test, like applying a chi-square test to a data set where the 'categories' are actually different statistical methods, would lead to incorrect conclusions. It's essential to correctly classify the data鈥攚hether it is indeed categorical or if it represents other types of information such as methodologies or frequencies鈥攂efore deciding on the appropriate statistical test.

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Most popular questions from this chapter

Which test, again? For each of the following situations, state whether you'd use a chi-square goodness-of-fit test, a chi-square test of homogeneity, a chi-square test of independence, or some other statistical test: a. Is the quality of a car affected by what day it was built? A car manufacturer examines a random sample of the warranty claims filed over the past two years to test whether defects are randomly distributed across days of the workweek. b. A medical researcher wants to know if blood cholesterol level is related to heart disease. She examines a database of 10,000 patients, testing whether the cholesterol level (in milligrams) is related to whether or not a person has heart disease. c. A student wants to find out whether political leaning (liberal, moderate, or conservative) is related to choice of major. He surveys 500 randomly chosen students and performs a test.

M\&M's As noted in an earlier chapter, Mars Inc. says that until very recently yellow candies made up \(20 \%\) of its milk chocolate M\&M's, red another \(20 \%,\) and orange, blue, and green \(10 \%\) each. The rest are brown. On his way home from work the day he was writing these exercises, one of the authors bought a bag of plain M\&M's. He got 29 yellow ones, 23 red, 12 orange, 14 blue, 8 green, and 20 brown. Is this sample consistent with the company's stated proportions? Test an appropriate hypothesis and state your conclusion. a. If the M\&M's are packaged in the stated proportions, how many of each color should the author have expected to get in his bag? b. To see if his bag was unusual, should he test goodness-of-fit, homogeneity, or independence? c. State the hypotheses. d. Check the conditions. e. How many degrees of freedom are there? f. Find \(\chi^{2}\) and the P-value. g. State a conclusion.

Human births If there is no seasonal effect on human births, we would expect equal numbers of children to be born in each season (winter, spring, summer, and fall). A student takes a census of her statistics class and finds that of the 120 students in the class, 25 were born in winter, 35 in spring, 32 in summer, and 28 in fall. She wonders if the excess in the spring is an indication that births are not uniform throughout the year. a. What is the expected number of births in each season if there is no "seasonal effect" on births? b. Compute the \(\chi^{2}\) statistic. c. How many degrees of freedom does the \(\chi^{2}\) statistic have?

Nuts A company says its premium mixture of nuts contains \(10 \%\) Brazil nuts, \(20 \%\) cashews, \(20 \%\) almonds, and \(10 \%\) hazelnuts, and the rest are peanuts. You buy a large can and separate the various kinds of nuts. On weighing them, you find there are 112 grams of Brazil nuts, 183 grams of cashews, 207 grams of almonds, 71 grams of hazelnuts, and 446 grams of peanuts. You wonder whether your mix is significantly different from what the company advertises. a. Explain why the chi-square goodness-of-fit test is not an appropriate way to find out. b. What might you do instead of weighing the nuts in order to use a \(\chi^{2}\) test?

Which test? For each of the following situations, state whether you'd use a chi-square goodness-of-fit test, a chisquare test of homogeneity, a chi-square test of independence, or some other statistical test: a. A brokerage firm wants to see whether the type of account a customer has (Silver, Gold, or Platinum) affects the type of trades that customer makes (in person, by phone, or on the Internet). It collects a random sample of trades made for its customers over the past year and performs a test. b. That brokerage firm also wants to know if the type of account affects the size of the account (in dollars). It performs a test to see if the mean size of the account is the same for the three account types. c. The academic research office at a large community college wants to see whether the distribution of courses chosen (Humanities, Social Science, or Science) is different for its residential and nonresidential students. It assembles last semester's data and performs a test.
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