Deck 10: The Analysis of Variance

Question 1

When is rejected in single-factor ANOVA, the investigator will usually want to know how much of the population means are different from one another. A method for carrying out this further analysis is called a __________.

Answer 1

multiple comparisons procedure

Question 2

In one-factor ANOVA, both mean square for treatments (MSTr) and mean square for error (MSE) are unbiased estimators for estimating the common population variance when __________, but MSTr tends to overestimate when __________.

Answer 2

is true, is false

Question 3

Let F =MSTr/MSE be the test statistic in a single-factor ANOVA problem involving four populations or treatments with a random sample of six observations from each one. When is true and the four population or treatment distributions are all normal with the same variance then F has an F distribution with degrees of freedom With f denoting the computed value of F, the rejection region for level .05 test is __________.

Answer 3

3, 20,

Question 4

In one-factor ANOVA, __________ denoted by __________ is the part of total variation that is unexplained by the truth or falsity of .

Question 5

In a single-factor ANOVA problem involving 5 populations, assume that the sample sizes from each population are not equal, and that the total number of observations is 21. If SSTr = 8 and SST = 12, then MSTr = __________, MSE = __________, and the test statistic value f = __________.

Question 6

Single-factor ANOVA focuses on a comparison of more than two population or treatment __________.

Question 7

Define a parameter by and the parameters by . Then, __________.

Question 8

An experiment is conducted to study the effects of the presence of four different sugar solutions (glucose, sucrose, fructose, and a mixture of the three) on bacterial growth. In this experiment, the factor of interest is __________, and there are __________ different levels of the factor.

Question 9

In single-factor ANOVA, SST - SSTr = __________.

Question 10

In a one-way ANOVA problem involving four populations or treatments, the null hypothesis of interest is

Question 11

In the "model equation" are assumed to be independent and normally distributed random variables with mean of 0 and standard deviation of for every i and j. Then the mean and variance of are __________ and __________, respectively.

Question 12

Sometimes an experiment is carried out to compare each of several new treatments to a control treatment. In such situations, a multiple comparisons technique called __________'s method is appropriate.

Question 13

In single-factor ANOVA, the __________ is a measure of between samples variation, and is denoted by __________.

Question 14

In one-factor ANOVA, __________ denoted by __________ is the part of total variation that can be explained by possible differences in the population means.

Question 15

In a single-factor ANOVA, the characteristic that differentiates the treatments or populations from one another is called the __________ under study, and the different treatments or populations are referred to as the __________ of the factor.

Question 16

If three 95% confidence intervals for a population mean are calculated based on three independent samples selected randomly from the population, then the simultaneous confidence level will be about __________%.

Question 17

The simplest ANOVA problem is referred to variously as a single-factor, single-classification, or __________ ANOVA.

Question 18

__________ multiple comparisons procedure involves the use of probability distributed called the Studentized range distribution.

Question 19

In single-factor ANOVA, the __________ is a measure of within-samples variation, and is denoted by __________.

Question 20

An experiment is conducted to study the effectiveness of three teaching methods on student performance. In this experiment, the factor of interest is __________, and there are __________ different levels of the factor.

Question 21

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.

Question 22

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.

Question 23

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.

Question 24

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.

Question 25

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.

Question 26

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

Question 27

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.

Question 28

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

Question 29

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.

Question 30

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.

Question 31

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.

Question 32

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.

Question 33

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.

Question 34

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

Question 35

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.

Question 36

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.

Question 37

If , a known function of , then a transformation that "stabilizes the variance" so that is approximately the same for each I is given by h(x)oc __________.

Question 38

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.

Question 39

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

Question 40

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.

Question 41

In a single-factor ANOVA problem involving 4 populations, the sample sizes are 7,5,6, and 6. If SST = 65.27 and SSTr = 23.49, then the test statistic value f is

A) 3.75
B) 2.09
C) 7.83
D) 0.56
E) 6.67

Question 42

Folacin is the only B vitamin present in any significant amount in tea, and recent advances in assay methods have made accurate determination of folacin content feasible. Consider the accompanying data on folacin content for randomly selected specimens of the four leading brands of green tea. Brand Observations $\begin{array} { l | l l l l l l l } \hline 1 & 8.0 & 6.3 & 6.7 & 8.7 & 9.0 & 10.2 & 9.7 \\2 & 5.8 & 7.6 & 9.9 & 6.2 & 8.5 & & \\3 & 6.9 & 7.6 & 5.1 & 7.5 & 5.4 & 6.2 & \\4 & 6.5 & 7.2 & 8.0 & 4.6 & 5.1 & 4.1 & \\\hline\end{array}$ Does this data suggest that true average folacin content is the same for all brands?

A) Carry out a test using $$\alpha = .05$$
Via the P-value method.
B) Perform a multiple comparisons analysis to identify significant differences among brands.

Question 43

Consider the accompanying data on plant growth after the application of different types of growth hormone. $$\begin{array} { l c | c c c c } \hline & 1 & 15 & 19 & 9 & 16 \\& 2 & 23 & 15 & 22 & 19 \\\text { Hormone } & 3 & 20 & 17 & 22 & 19 \\& 4 & 9 & 13 & 20 & 12 \\& 5 & 8 & 13 & 17 & 10 \\\hline\end{array}$$

A) Perform an F test at level $\alpha = .05$
B) What happens when Tukey's procedure is applied?