1. Tail-feather length in birds is sometimes a sexually dimorphic trait.
That is, the trait differs substantially for males and for females.
Researchers measured tail-feather length (the R1 central tail feather, in mm) in male and female longtailed finches either raised in an aviary or caught in the wild.
This observational study does not have a balanced design, particularly because finches caught in the wild were harder to obtain.
A total of 52 finches were studied.
A partial ANOVA table is provided below,
Source DF SS MS F P
Gender 1 2015.5 2015.5
origin 1 251.7 251.7
interaction 1 15.6 15.6
Error 48 4076.9 84.9
S = 9.21600 R-Sq = 47.32% R-Sq(adj) = 44.03%
a) What is the data model for the study?
b) Complete the ANOVA table and answer the following questions
The test statistic for the interaction effect is ________________, p-value_________;
The test statistic for the main gender effect is ________________, p-value_________;
The test statistic for the main origin effect is ________________, p-value_________;
c) When discuss the interaction effect with a significant level of 0.05, we see there is _____
A) a non-significant interaction effect, therefore we can discuss the main effects separately.
B) a non-significant interaction effect, therefore sex and origin do not influence mean tail-feather length in finches.
C) a significant interaction effect, therefore we can discuss the main effects separately.
D) a significant interaction effect, therefore we should discuss the main effects together.
2. An outbreak of the deadly Ebola virus in 2002 and 2003 killed 91 of the 95 gorillas in 7 home ranges in the Congo. To study the spread of the virus, measure "distance" by the number of home ranges separating a group of gorillas from the first group infected.
Here are data on distance and number of days until deaths began in each later group:2
a).Make a scatterplot with distance as the explanatory variable. Judge from the scatter plot and tell whether there is any association between them?
b) Compute the correlation coefficient. Show all details. Judge from the correlation coefficient and tell whether there is any LINEAR association?
c) Find the linear regression function, and the 95% confidence interval for the linear impact (slope).
d) Calculate by hand the residuals for the six data points.
Is their sum 0 (aside from round off error)? Use the residuals to estimate the standard deviation σ that measures variation in the responses (days) about the means given by the population regression line, in other words, denote and compute the residual standard error.
e) Make a residual plot then check assumptions for the linear regression model.
3. Refer to problem 2, which gives data on the distance of gorilla groups from the origin of an Ebola outbreak and the number of days until deaths from Ebola began.
Software tells us that the least-squares slope is b = 11.263 with standard error SEb = 1.591.
a) Perform a hypothesis to address the problem that the distance has a significant positive linear impact on the days. Choose and define the hypothesis with the right notation.
b) Perform a hypothesis to address the problem that the distance and days have a significant linear correlation. Choose anddefine the hypothesis with the right notation.
c) Compare a) and b). What is the difference between linear impact and linear correlation? Sketch a scatter plot (plot I) where there is a small linear impact but big linear correlation; then sketch a second scatter plot (plot II) where there is a big linear impact but small linear correlation.
4. Refer to problem 3, find a 95% Confidence interval for
a) Predicting the mean response of days given distance is 5
b) Predicting a single observation of days given distance is 5.