ISYE 6414 - Midterm 1 Prep Exam
Questions with Correct Answers
In ANOVA, the primary objectives are to: - Answer-1) Analyze the variability in the data
using the ANOVA table
2) Use this analysis of variance
3) Estimate confidence intervals
Analyze the variability in the data using the ANOVA table - Answer-That means we
compare the variability within each group to the variability between the means. We
represent all the information in a table, laying out all the components needed to make
the comparison.
Testing for equal means - Answer-Specifically, we will test the null hypothesis that all
means are equal versus the alternative that at least two of the means are not equal.
Estimate confidence intervals - Answer-for all the pairs of means, in order to identify
which of the means are not equal, or which of the means are statistically significantly
different. Specifically, we will consider a hypothesis test for each pair of means with the
null hypothesis that the means in the pair are equal versus the alternative that are not
equal. We will perform all the hypothesis tests across all pair jointly
j - Answer-is the index within group
i - Answer-is the index across groups
pooled variance estimator (Si^2) - Answer-are the sample variances of the data
samples
of the response variable. The formula is also called the mean square error in
ANOVA or MSE.
N-k - Answer-The degree of freedom in the estimation in the pooled variance estimator
the sampling distribution of the pooled variance - Answer-is a chi-square
distribution with N - k degrees of freedom (t distribution)
confidence intervals for the Means - Answer-we can use the estimated sample means
and the estimated variance to calculate (1-α) confidence intervals for the treatment
means.
aov() - Answer-R command to fit an ANOVA model using the R statistical software
, model.table() - Answer-R command to get the estimated means
The null hypothesis for equal means - Answer-mu 1 = mu 2...=mu k
alternative hypothesis - Answer-some means are different. Not all means have to be
different for the alternative hypothesis to be true -- at least one pair of the means needs
to be different.
N - Answer-is the sum of all samples
If we want to estimate the variance - Answer-similarly we're going to use the sample
variance of the entire combined response samples.
SST / N - 1.
Because we are replacing only one parameter, the overall mean, we're now only losing
one degree of freedom.
The difference between this variance estimator and the pool variance estimator -
Answer-is that now we are replacing the mean with the overall mean, not with the
individual sample means
SST - Answer-SSE + SSTR (sum of squared error plus the sum of square of
treatments)
SSE - Answer-is the sum of square differences between the observations and the
individual sample means
SSTR - Answer-is the sum of the square difference between the sample means of the
individual samples minus the overall mean
MSE (ANOVA) - Answer-SSE/N-k = within-group variability
MSST - Answer-SST/k-1 = between-group variability
ANOVA - Answer-comparing between to within variability
F-test - Answer-is the ration of between-group variability and withing-group variability
If the F-test value is large, - Answer-variability between groups is larger than variability
within
groups, and thus we reject the null hypothesis that the means are equal.
pairwise comparison - Answer-to determine which treatment means are
bigger or smaller. One way to do this is to compare all possible pairs; there are k(k-1)/2
unique pairs of treatments
q critical point - Answer-allows for
Questions with Correct Answers
In ANOVA, the primary objectives are to: - Answer-1) Analyze the variability in the data
using the ANOVA table
2) Use this analysis of variance
3) Estimate confidence intervals
Analyze the variability in the data using the ANOVA table - Answer-That means we
compare the variability within each group to the variability between the means. We
represent all the information in a table, laying out all the components needed to make
the comparison.
Testing for equal means - Answer-Specifically, we will test the null hypothesis that all
means are equal versus the alternative that at least two of the means are not equal.
Estimate confidence intervals - Answer-for all the pairs of means, in order to identify
which of the means are not equal, or which of the means are statistically significantly
different. Specifically, we will consider a hypothesis test for each pair of means with the
null hypothesis that the means in the pair are equal versus the alternative that are not
equal. We will perform all the hypothesis tests across all pair jointly
j - Answer-is the index within group
i - Answer-is the index across groups
pooled variance estimator (Si^2) - Answer-are the sample variances of the data
samples
of the response variable. The formula is also called the mean square error in
ANOVA or MSE.
N-k - Answer-The degree of freedom in the estimation in the pooled variance estimator
the sampling distribution of the pooled variance - Answer-is a chi-square
distribution with N - k degrees of freedom (t distribution)
confidence intervals for the Means - Answer-we can use the estimated sample means
and the estimated variance to calculate (1-α) confidence intervals for the treatment
means.
aov() - Answer-R command to fit an ANOVA model using the R statistical software
, model.table() - Answer-R command to get the estimated means
The null hypothesis for equal means - Answer-mu 1 = mu 2...=mu k
alternative hypothesis - Answer-some means are different. Not all means have to be
different for the alternative hypothesis to be true -- at least one pair of the means needs
to be different.
N - Answer-is the sum of all samples
If we want to estimate the variance - Answer-similarly we're going to use the sample
variance of the entire combined response samples.
SST / N - 1.
Because we are replacing only one parameter, the overall mean, we're now only losing
one degree of freedom.
The difference between this variance estimator and the pool variance estimator -
Answer-is that now we are replacing the mean with the overall mean, not with the
individual sample means
SST - Answer-SSE + SSTR (sum of squared error plus the sum of square of
treatments)
SSE - Answer-is the sum of square differences between the observations and the
individual sample means
SSTR - Answer-is the sum of the square difference between the sample means of the
individual samples minus the overall mean
MSE (ANOVA) - Answer-SSE/N-k = within-group variability
MSST - Answer-SST/k-1 = between-group variability
ANOVA - Answer-comparing between to within variability
F-test - Answer-is the ration of between-group variability and withing-group variability
If the F-test value is large, - Answer-variability between groups is larger than variability
within
groups, and thus we reject the null hypothesis that the means are equal.
pairwise comparison - Answer-to determine which treatment means are
bigger or smaller. One way to do this is to compare all possible pairs; there are k(k-1)/2
unique pairs of treatments
q critical point - Answer-allows for
