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Chapter 27: Problem 21

Which of the following would be evidence that ANOVA inference is not safe to use? a. The sample sizes are less than four. b. The largest sample variance is more than twice as large as the smallest sample variance. c. The largest sample standard deviation is more than twice as large as the smallest sample standard deviation.

Short Answer

Expert verified
Option c: The largest sample standard deviation is more than twice as large as the smallest.

Step by step solution

01

Understanding ANOVA Assumptions

ANOVA (Analysis of Variance) requires certain assumptions to be valid. These include: independence of observations, normality of the distributions, and homogeneity of variances (equal variances across groups). If these assumptions are violated, the results from ANOVA may not be reliable.
02

Assess Sample Size

ANOVA requires a sufficient sample size to detect differences among group means. Having less than 4 observations in a group (Option a) is generally too small, making the inference potentially unreliable. However, this is a general guideline and not an absolute rule against ANOVA usage if other conditions are satisfied.
03

Evaluate Variance Homogeneity

Equal variances across groups is a key assumption of ANOVA. If the largest sample variance is more than twice as large as the smallest sample variance (Option b), it indicates a potential violation of this assumption, which can compromise the reliability of ANOVA results.
04

Examine Standard Deviation Ratio

Standard deviation is the square root of variance. If the largest sample standard deviation is more than twice as large as the smallest sample standard deviation (Option c), it also suggests a violation of the equal variance assumption, which ANOVA relies upon to compare group means accurately.
05

Conclude Evidence Against ANOVA

Both variance and standard deviation provide information on the spread of data. Particularly, having the largest standard deviation more than twice the smallest (Option c) highlights significant variance heterogeneity, offering clear evidence against safely using ANOVA inference.

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Key Concepts

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

Sample Size
In the context of ANOVA, or Analysis of Variance, sample size is a crucial factor that can influence the reliability of the test results. ANOVA is designed to compare the means of different groups and requires a sufficient number of observations in each group to produce meaningful results. Typically, the guideline is to have at least four observations per group. When the sample size falls below this number, the test might not have enough power to detect actual differences between the groups鈥� means. A smaller sample size can lead to higher variability and less accurate estimations of group variances.
Therefore, the inferences drawn might become less reliable. Consider this:
  • A small sample size increases the margin of error.
  • It may lead to larger confidence intervals.
  • There is an increased risk of Type II error, meaning real differences might go undetected.
Thus, ensuring a sufficiently large sample size is fundamental when planning to use ANOVA.
Variance Homogeneity
Homogeneity of variances, also known as homoscedasticity, is one of the pivotal assumptions in conducting an ANOVA test. ANOVA relies on the assumption that the variances in each group being compared are equal. This allows for a proper comparison of means across the groups without biases from unequal variance. But why is this important? When variances are equal, it implies that the groups have similar variability, helping to ensure that any observed differences in means are due to true differences and not due to variation in the data spread. Signs of variance heterogeneity include:
  • The largest sample variance being more than twice the smallest sample variance.
  • Significantly different spread in data within groups.
When this assumption is violated, the results of the ANOVA test may not be trustworthy, possibly leading to incorrect conclusions about the data.
Standard Deviation
Closely related to variance is the standard deviation, which is another measure of data spread in a dataset. It's defined as the square root of variance and depicts how much individual data points differ from the mean of the dataset. For ANOVA to function correctly, the standard deviations among groups should not be drastically different. Violation scenarios include:
  • The largest standard deviation is more than twice as large as the smallest standard deviation.
  • Unequal data spread which may indicate underlying variance issues.
If such discrepancies are observed, it indicates that the assumption of equal variance is breached.
This can significantly alter ANOVA results, making conclusions unreliable and potentially misleading, as the variance component heavily influences the ANOVA test statistic.

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Most popular questions from this chapter

What Music Will You Play? People often match their behavior to their social environment. One study of this idea first established that the type of music most preferred by Black college students is R\&B and that Whites' most preferred music is rock. Will students hosting a small group of other students choose music that matches the racial composition of the people attending? Assign 90 Black business students at random to three equal-sized groups. Do the same for 96 White students. Each student sees a picture of the people he or she will host. Group 1 sees six Blacks, Group 2 sees three Whites and three Blacks, and Group 3 sees six Whites. Ask how likely the host is to play the type of music preferred by the other race. Use ANOVA to compare the three groups to see whether the racial mix of the gathering affects the choice of music. \(\frac{8}{1}\) a. For the White subjects, \(F=16.48\). What are the degrees of freedom? b. For the Black subjects, \(F=2.47\). What are the degrees of freedom?

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Vitamin \(\mathrm{D}\) is important for maintaining strong bones by helping the body absorb calcium from food and supplements. The recommended amount of vitamin D is 400 IU (international units) per day. Many people take vitamin D supplements, often at levels greatly exceeding the recommended daily amount. What is the effect of taking too much vitamin D? Researchers investigated this in a double-blind randomized clinical trial. Subjects were divided into three groups. One, consisting of 109 subjects, took 400 IU per day for three years. A second group, consisting of 100 subjects, took 4,000 IU per day for three years. A third group, consisting of 102 subjects, took 10,000 IU per day for three years. The total volumetric bone mass density of the tibia (the large bone located in the lower front portion of the leg, sometimes referred to as the shinbone) was measured in order to assess the effect of the different dosages of vitamin D. \(\underline{11}\) The degrees of freedom for the ANOVA \(F\) statistic comparing the mean volumetric bone masses are a. 2 and \(98 .\) b. 2 and 308 . c. 3 and 310 .

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