Question 1

Which of the following statements are not true?
A) The analysis of variance, or more briefly ANOVA, refers broadly to a collection of experimental situations and statistical procedures for the analysis of quantitative responses from experimental units.
B) The simplest ANOVA problem is referred to as two-way ANOVA.
C) Single-factor ANOVA focuses on a comparison of more than two population or treatment means.
D) All of the above statements are true.
E) None of the above statements are true.

Accepted Answer

The simplest ANOVA problem is referred to as one-way ANOVA, not two-way ANOVA.

Question 2

The distribution of the test statistic in single-factor ANOVA is the
A) binomial distribution
B) normal distribution
C) t distribution
D) F distribution
E) None of the above answers are correct.

Accepted Answer

The distribution of the test statistic in single-factor ANOVA is the F distribution. This is because ANOVA involves comparing the variation between groups to the variation within groups. The F statistic calculates the ratio of these two variances, and its distribution follows the F distribution.

Question 3

If three 90% confidence intervals for a population proportion p are calculated based on three independent samples selected randomly from the population, then the simultaneous confidence level will be
A) about 73%
B) exactly 90%
C) exactly 81%
D) exactly 270%
E) None of the above answers are correct.

Accepted Answer

The simultaneous confidence level will be lower than any single confidence level because there is greater variability when multiple intervals are considered. One way to approximate the simultaneous confidence level is to use the Bonferroni correction, which divides the desired confidence level by the number of intervals being considered. In this case, since there are three intervals, the Bonferroni correction gives a confidence level of 0.90/3 = 0.30. To convert to a percentage, we multiply by 100 to get 30%. However, since this is the lower bound on the confidence level, we can say that the simultaneous confidence level is about 73% (which is closer to choice A than choice C).

Question 4

Which of the following statements are not true?
A) An F distribution arises in connection with a ratio in which there is one number of degrees of freedom associated with the numerator $\left( v _ { 1 } \right)$
And another number of degrees of freedom associated with the denominator $\left( v _ { 2 } \right)$
)
B) For single-factor ANOVA, a value of the test statistic F = MSTr/MSE that is greatly smaller than 1, casts considerable doubt on $H _ { o }$
)
C) The unbiasedness of MSE in single-factor ANOVA is a consequence of $E \left( S _ { i} ^ { 2 } \right) = \sigma ^ { 2 }$
(the common population variance) whether $H _ { o }$
Is true or false.
D) In a single-factor ANOVA problem, the F critical value that captures upper-tail area .05 under the F curve with $v _ { 1 } = 6 \text { and } v _ { 2 } = 4 \text { is } 6.16$
E) All of the above statements are true.

Accepted Answer

The answer of Which of the following statements are not...

Question 5

Consider calculating a 95% confidence interval for a population mean $\mu$ based on a sample from a population, and then a 95% confidence interval for a population proportion p based on another sample selected independently from the same population. Which of the following statements are true?
A) Prior to obtaining data, the probability that the first interval will include $\mu$
Is )95.
B) Prior to obtaining data, the probability that the second interval will include p is .95.
C) The probability that both intervals will include the values of the respective parameters is about .90.
D) All of the above statements are correct.

Accepted Answer

Each statement reflects the correct interpretation of confidence intervals. Prior to obtaining data, the probability that a 95% confidence interval will include the true parameter (mean $\mu$ or proportion $p$) is 0.95. The probability that both intervals include their respective true parameters is not directly calculated as a simple product (which would suggest 0.9025), but the conceptual understanding of confidence intervals supports the idea that each interval is constructed to include its true parameter 95% of the time, hence all statements are considered correct in the context of confidence interval interpretation.

Question 6

In a single-factor ANOVA problem involving 3 treatments, the sample means were 5,6, and 9. If each observation in the third sample was increased by 20, the test statistic value f would
A) increase
B) decrease
C) remain the same
D) increase by 20
E) decrease by 10

Accepted Answer

Increasing each observation in the third sample by 20 will increase the variance between the sample means, leading to an increase in the F statistic.

Question 7

Which of the following statements are true?
A) In some experiments, different samples contain different numbers of observations. However, the concepts and methods of single-factor ANOVA are most easily developed for the case of equal sample sizes.
B) The population or treatment distributions in single-factor ANOVA are all assumed to be normally distributed with the same variance $\sigma ^ { 2 }$
C) In one-way ANOVA, if either the normality assumption or the assumption of equal variances is judged implausible, a method of analysis other than the usual F test must be employed.
D) The test statistic for single-factor ANOVA is F = MSTr/MSE, where MSTr is the mean square for treatments, and MSE is the mean square for error. 
E) All of the above statements are true.

Accepted Answer

The answer of Which of the following statements are true?
A)...

Question 8

In a single-factor ANOVA problem involving five populations or treatments with a random sample of four observations form each one, it is found that SSTr = 16.1408 and SSE = 37.3801. Then the value of the test statistic is
A) 1.619
B) 2.316
C) 0.432
D) 1.522
E) 4.248

Accepted Answer

The test statistic for ANOVA is calculated as MSTr/MSE. MSTr = SSTr/(k-1) = 16.1408/4 = 4.0352. MSE = SSE/(N-k) = 37.3801/15 = 2.492. The test statistic = 4.0352/2.492 = 1.619.

Question 9

Which of the following statements are not true?
A) When the computed value of the F statistic in single-factor ANOVA is significant, the analysis is terminated because no differences among the population means have been identified.
B) When $H _ { o }$
Is rejected in single-factor ANOVA, further analysis is carried out by applying the multiple comparisons procedure.
C) Tukey's multiple comparisons procedure involves the use of probability distribution called the Studentized range distribution.
D) All of the above statements are true.
E) None of the above statements are true.

Accepted Answer

EA) This statement is not true because when the F statistic in ANOVA is significant, it indicates that there are differences among the population means, not that there are no differences. The analysis does not terminate; instead, further analysis may be needed to identify specific group differences.E) Given that statement A is not true, option E is also correct because it states that none of the above statements are true, which applies to statement A.

Question 10

In a single-factor ANOVA problem involving five populations or treatments, which of the following statements are true about the alternative hypothesis?
A) All five population means are equal.
B) All five population means are different.
C) At least two of the population mean are different.
D) At least three of the population mean are different.
E) At most, two of the population means are equal.

Accepted Answer

The alternative hypothesis for a single-factor ANOVA problem involving five populations or treatments is that at least two of the population means are different. This is because we are not testing whether all five population means are equal (that is the null hypothesis), but rather whether there is enough evidence to suggest that at least two of them are different.

Question 11

Which of the following statements are not true? A) The two-sample t test is more flexible than the F test when the number of treatments or populations is 2. B) The two-sample t test is valid without the assumption that the two population variances are equal. C) The two-sample t test can be used to test $H _ { \pm } : \mu _ { 1 } > \mu _ { 2 } \text { or } H _ { \pm} : \mu _ { 1 } < \mu _ { 2 }$ As well as $H _ { \pm } : \mu _ { 1 } \neq \mu _ { 2 }$ ) D) The F test can be used to test $H _ { \pm } : \mu _ { 1 } > \mu _ { 2 } \text { or } H _ { \pm } : \mu _ { 1 } < \mu _ { 2 }$ As well as $H _ { \pm} : \mu _ { 1 } \neq \mu _ { 2 }$ ) E) When the number of treatments or populations is at least 3, there is no general test procedure known to have good properties without assuming equal populations variances.

Accepted Answer

The answer of Which of the following statements are not...

Question 12

In one-way ANOVA, which of the following statements are true?
A) SST is a measure of the total variation in the data.
B) SSE measures variation that would be present within treatments even if $H _ { o }$
Were true, and is thus the part of total variation that is unexplained by the truth or falsity of $H _ { o }$
)
C) SSTr is the amount of variation between treatments that can be explained by possible differences in the population or treatments' means.
D) If explained variation is large relative to unexplained variation, then $H _ { o }$
Is rejected in favor of $H _ { \pm }$
)
E) All of the above statements are true.

Accepted Answer

All of the statements accurately describe aspects of one-way ANOVA. A) SST represents the total variation in the dataset. B) SSE is the error sum of squares, indicating variation within groups that is not explained by the treatment effects. C) SSTr measures the variation between group means, which could be due to treatment effects. D) A large ratio of explained to unexplained variation (SSTr/SSE) suggests that the null hypothesis ($H_0$) should be rejected, indicating significant treatment effects.

Question 13

Which of the following statements are not correct?
A) The simultaneous confidence level is controlled by Tukey's method.
B) The Tukey intervals are based on independent samples.
C) To obtain a 95% simultaneous confidence level using the Tukey method, the individual level for each interval must be considerably larger than 95%.
D) All of the above statements are true.
E) None of the above statements are true.

Accepted Answer

Tukey's method is used for multiple comparisons of means within the same set of data, implying that the samples are not independent but rather related or paired as part of the same experiment or study.

Question 14

In single-factor ANOVA, MSTr is the mean square for treatments, and MSE is the mean square for error. Which of the following statements are not true?
A) MSE is a measure of between-samples variation.
B) MSE is a measure of within-samples variation.
C) MSTr is a measure of between-samples variation.
D) The value of MSTr is affected by the status of $H _ { o }$
(true or false).
E) All of the above statements are true

Accepted Answer

The answer of In single-factor ANOVA, MSTr is the mean...

Question 15

In single-factor ANOVA, MSE is the mean square for error, and MSTr is the mean square for treatments. Which of the following statements are not true?
A) The value of MSTr is affected by the status of $H _ { o }$
(true or false)
B) When $H _ { o }$
Is true, E(MSTr) = E(MSE) = $\sigma ^ { 2 } \text { where } \sigma ^ { 2 }$
Is the common population variance.
C) When $H _ { o }$
Is false, E(MSTr) > E(MSE) = $\sigma ^ { 2 } \text { where } \sigma ^ { 2 }$
Is the common population variance.
D) The value of MSE is affected by the status of $H _ { o }$
(true or false).
E) All of the above statements are true.

Accepted Answer

The answer of In single-factor ANOVA, MSE is the mean...

Question 16

The assumptions of single-factor ANOVA can be described succinctly by means of the "model equation" $X _ {ij } = \mu _ { i } + \varepsilon _ {  ij } \text { where } \varepsilon _ { ij }$ represents a random deviation from the population or true treatment mean $\mu _ { i }$ . Which of the following statements are true?
A) The $\varepsilon _ {ij } ^ { \prime } \mathrm { s }$
Are assumed to be independent.
B) The $\varepsilon _ { i j } ^ { \prime } \mathrm { s }$
Are normally distributed random variables.
C) $E \left( \varepsilon _ { ij } \right) = 0$
For every i and j.
D) $V \left( \varepsilon _ {ij } \right) = \sigma ^ { 2 }$
For every i and j.
E) All of the above statements are true.

Accepted Answer

All of the statements A through D accurately describe the assumptions of single-factor ANOVA. The errors ($\varepsilon_{ij}$) are assumed to be independent, normally distributed with a mean of 0, and have a constant variance ($\sigma^2$) across all groups.

Question 17

If $V \left( X _ { y } \right) = g \left( \mu _ { i } \right)$ , a known function of $\mu$ , then a transformation $h \left( X _ { y } \right)$ that "stabilizes the variance" so that $V \left[ h \left( X _ { y } \right) \right]$ is approximately the same for each I is given by h(x)oc __________.

Accepted Answer

The answer of If \(V \left( X _ { y...

Question 18

In a single-factor ANOVA problem involving four populations or treatments, the four sample standard deviations are 25.6, 30.4, 28.7, and 32.50. Then, the mean square for error is
A) 29.3
B) 117.2
C) 864.865
D) 29.409
E) None of the above answers are correct.

Accepted Answer

The answer of In a single-factor ANOVA problem involving four...

Question 19

In a single-factor ANOVA problem involving five populations or treatments with a random sample of nine observations from each one, suppose that $H _ { o }$ is rejected at .05 level. Which of the following values are correct for the appropriate $Q _ { \alpha }$ critical value needed to perform Tukey's procedure?
A) 4.76
B) 3.79
C) 4.04
D) 3.85
E) 4.80

Accepted Answer

The correct critical value for Tukey's procedure can be found using the studentized range distribution table for $ \alpha = 0.05 $, $ k = 5 $ groups, and $ df = 40 $ (since $ n = 9 $ for each group, total $ N = 45 $, and $ df = N - k = 40 $). The closest value in the table is 4.04.

Question 20

Which of the following statements are not true?
A) ANOVA can be used to test $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \text { versus } H _ { \pm } : \mu _ { 1 } \neq \mu _ { 2 }$
)
B) ANOVA cannot be used to test $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \text { versus } H _ { \pm } : \mu _ { 1 } \neq \mu _ { 2 }$
)
C) ANOVA can be used to test $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } \text { versus } H _ { \pm } :$
At least two of the $\mu _ { i } ^ { \prime } s$
Are different.
D) The two-sample t test can be used to test $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 } \text { versus } H _ { \pm } : \mu _ { 1 } \neq \mu _ { 2 }$
)
E) All of the above statements are true.

Accepted Answer

The answer of Which of the following statements are not...

Quiz 10: The Analysis of Variance

Which of the following statements are not true?

The distribution of the test statistic in single-factor ANOVA is the

If three 90% confidence intervals for a population proportion p are calculated based on three independent samples selected randomly from the population, then the simultaneous confidence level will be

Which of the following statements are not true?

Consider calculating a 95% confidence interval for a population mean μ\muμ based on a sample from a population, and then a 95% confidence interval for a population proportion p based on another sample selected independently from the same population. Which of the following statements are true?

In a single-factor ANOVA problem involving 3 treatments, the sample means were 5,6, and 9. If each observation in the third sample was increased by 20, the test statistic value f would

Which of the following statements are true?

In a single-factor ANOVA problem involving five populations or treatments with a random sample of four observations form each one, it is found that SSTr = 16.1408 and SSE = 37.3801. Then the value of the test statistic is

Which of the following statements are not true?

In a single-factor ANOVA problem involving five populations or treatments, which of the following statements are true about the alternative hypothesis?

Which of the following statements are not true?

In one-way ANOVA, which of the following statements are true?

Which of the following statements are not correct?

In single-factor ANOVA, MSTr is the mean square for treatments, and MSE is the mean square for error. Which of the following statements are not true?

In single-factor ANOVA, MSE is the mean square for error, and MSTr is the mean square for treatments. Which of the following statements are not true?

In a single-factor ANOVA problem involving four populations or treatments, the four sample standard deviations are 25.6, 30.4, 28.7, and 32.50. Then, the mean square for error is

Which of the following statements are not true?

Consider calculating a 95% confidence interval for a population mean $\mu$ based on a sample from a population, and then a 95% confidence interval for a population proportion p based on another sample selected independently from the same population. Which of the following statements are true?