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Statistics 101A
Homework Two Due Friday Jan. 31st 2020 @ 5 pm
Question One: Problem one from chapter three 3.4 Exercises (The data file airfares.txt)
Question Two: Problem eight from chapter three 3.4 exercises (The Diamond stones
data file)
Question Three:
a) Using the stress echo UCLA data (see week four), fit a linear model to predict
basal blood pressure from systolic blood pressure. Report the equation for the
model. Report a residual plot and comment what it tells us about the assumption
of linearity.
b) Report the ANOVA table. Show how you can find the F value reported in the
ANOVA table using R2
. What is the null hypothesis that you are testing through
ANOVA? Compare the F value that you calculate with value that you find from
the F table and decide whether you are going to reject or fail to reject the null
hypothesis). Check if this equation is true: (𝑆𝑆𝑆𝑆)2 is approximately equal to
𝑣𝑣𝑣𝑣𝑣𝑣(𝑌𝑌) ∗ (1 – 𝑟𝑟2).
c) Calculate R2 adjusted and compare it to R2
. Comment on the difference.
d) Check the diagnostic plots and comment on each one of them.
e) Create two new variables: one for the leverage of a point and one for the
standardized residuals. Create a table from both variables to identify the
following:
Leverage/Outliers Yes No
Yes
No
f) Use ggplot2 library to create a plot of Leverage Vs Standardizes residuals
divided into regions to help you identify bad and good leverage points, outliers
and not leverage points and all the ordinary points.
Question Four:
Use the Echo data from question three to transform the data and compare the results to
the SLR created in question three:
a) Use the inverse response plot to find the best λ to transform the y variable to
minimize the SSE. Construct a SLR of the transformed y variable and systolic
blood pressure. Check diagnostics. Is this one better than the SLR in question
three.
b) Use the power transform function to find the best λ(s) to transform both the y
variable and the x variable to make the densities of these two variables as close as
possible to normal. Construct a SLR of the transformed variables. Check
diagnostics. Is this one better than the SLR in question three.
Question Five:
Consider the following R output predicting Marine water growth from Freshwater growth
in Salmon:
> SL1<- lm(salmon$Marine~salmon$Freshwater)
> summary(SL1)
Call:
lm(formula = salmon$Marine ~ salmon$Freshwater)
Residuals:
Min 1Q Median 3Q Max
-88.222 -27.382 -3.406 24.784 89.977
Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 511.3656 18.2547 28.01 < 2e-16 ***
salmon$Freshwater -0.9602 0.1512 -6.35 6.75e-09 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 39.12 on 98 degrees of freedom
Multiple R-squared: 0.2915, Adjusted R-squared: 0.2843
F-statistic: 40.32 on 1 and 98 DF, p-value: 6.747e-09
> summary(salmon$Freshwater)
Min. 1st Qu. Median Mean 3rd Qu. Max. 53.0 99.0 117.5 117.9 140.0 179.0
> var(salmon$Freshwater)
[1] 676.0541
> summary(salmon$Marine)
Min. 1st Qu. Median Mean 3rd Qu. Max. 301.0 367.0 396.5 398.1 428.2 511.0
> var(salmon$Marine)
[1] 2138.142
a) Construct ANOVA table based on the given output.
Consider the three observations: 4, 41 and 53 Observation SalmonOrigin Freshwater Marine
1 4 Alaska 86 506
2 41 Alaska 84 511
3 53 Canada 179 407
b) Which of these three points is(are) a leverage point?
c)Which of these three points is (are) an outlier?
d) Based on your answers of part b and c, classify these points as one of the
following:
i) A bad leverage point ii) An outlier but Not a leverage point.
iii) A good leverage point iv) Not a leverage point nor an outlier (ordinary)

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