Stat 503 Homework 13 (50pt) Due Thursday 12/10

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[Learning process checkup] This week we focus on correlation and linear regression.

Lecture recording: Topic 17

[Review question 1] Sketch two scatter plots, where one shows (approximately) r , and the other shows .

[Review question 2] Explain the difference between the usual sample standard deviation (denoted by s) and the residual standard deviation (denoted by s(resid)).

[Review question 3] Interpret the following output of a regression analysis.

Regression Analysis:

The regression equation is

Frequency = - 6.19 + 2.33 Temperature

Predictor Coef SE Coef T P

Constant -6.190 8.243 -0.75 0.462

Temperature 2.3308 0.3468 6.72 0.000

S = 2.82159 R-Sq = 71.5% R-Sq(adj) = 69.9%

Use the information provided to calculate the 95% CI for the linear slope. Interpret your result in terms of linear effect.

1. Tail-feather length in birds is sometimes a sexually dimorphic trait. That is, the trait differs substantially for males and females. Researchers measured tail-feather length (the central tail feather, in mm) in male and female long-tailed finches either raised in an aviary or caught in the wild. 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) 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_________;

b) When we examine the interaction effect with a significant level of 0.05, we see there is _____

B) a non-significant interaction effect, therefore we can study the main effects separately.

C) a non-significant interaction effect, therefore sex and origin do not influence mean tail-feather length in finches.

D) a significant interaction effect, therefore we can discuss the main effects separately.

E) 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 seven home ranges in the Congo. To study the spread of the virus, researchers 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:

a) Make a scatterplot with Distance as the explanatory variable and Days as the response variable. Judge from the scatter plot whether there is any association between them.

b) Compute the correlation coefficient. Judge from the correlation coefficient whether there is any LINEAR association.

c) Find the linear regression function, and the 95% confidence interval for the slope.

d) Calculate by hand the residuals for the six data points. Do they sum to 0 (aside from round off error)?

e) Compute the residual standard deviation, which estimates the standard deviation σ that measures variation in the responses (days) about the means given by the population regression line.

f) Make a residual plot then check assumptions for the linear regression model.

3. Refer to Question 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 test to address whether Distance has a significant positive linear impact on Days. Define the hypotheses with the right notation.

b) Perform a hypothesis test to address whether Distance and Days are linearly correlated. Define the hypotheses with the right notation.

4. [OQ]. Create your OWN linear regression problem. Make a data set for your own context and then use excel to address the following items.

a) Use a hypothesis test to assess linear correlation between the two variables.

b) Use a hypothesis test to examine a positive linear slope.

c) Use a confidence interval to estimate the linear impact.