Stat 230 Assignment 6 2020W2
Question 1
Suppose we have I samples (each of size J): X1, X2, · · · , XI , taken from each of I possibly
different populations. All observations are independent, inside and between the samples. The
table below shows information of each samples
Sample
Observation in
Each Sample
Question 2
(8 points)An experiment was conducted in which four brands of tires were compared for traction
on ice. Three measurements of stopping distance were taken for each type of tire.
In the experiment, a single car was used. In each experimental run, the distance over which
the car could decelerate from 20 km/h to 0 km/h on an artificial ice surface was measured.
Tire Brand Stopping Distance
Compute the treatment group averages and the residuals. Do the ANOVA and decide whether
the true mean stopping distances are different for the different brands at the .05 level.
Question 3
(5 points) The time in months before paint started to peel for five brands of paint applied to
a set of test panels is summarized below:
Paint No. of Panels ¯xi· s
2
A 6 48.6 30.5
B 6 51.2 30.2
C 6 60.1 31.8
D 6 55.2 33.1
On the basis of residual plots, the data can be modeled as normal, and the measurements
appear to be independent. Is there evidence at the 5% level of a difference in the mean time
to peel for the different panels?
Question 4
(4 points)The data set PlantGrowth is available in R. It contains weight measurements on
30 plants grown under 3 different experimental conditions (10 in each condition). Here is the
output from the aov command in R:
Df Sum Sq Mean Sq F value Pr(>F)
group 2 3.7663 1.8832 4.8461 0.01591 *
Residuals 27 10.4921 0.3886
Is there evidence that at least one experimental condition yields plants of a different mean
weight?
Question 5
An individual’s critical flicker frequency (cff) is the highest frequency (in cps) at which the
flicker in a flickering light source can still be detected. At frequencies above the cff, the light
source appears to be continuous even though it is actually flickering. A preliminary investigation
was conducted to see if true average cff depends on iris color. The results are given below from
an ANOVA carried out in R. An equal number of rats were fed one of three diets (Beef, Pork,
or Cereal) in order to investigate weight gain. The partial results from an ANOVA carried out
in R follow below.(Data sourced from the graonva package in R)
Df Sum Sq Mean Sq F value
Diet 267
Residuals 57 15932
(a) (1 point) How many rats were used in this study?
(b) (4 points) Complete this ANOVA table and interpret the result of the associated F-test.
Question 6
(8 points) Suppose x1, x2, . . . , xn come from a Poisson distribution p(x) = e−λλ
x
x!
, where x =
0, 1, 2, . . . with unknown parameter λ. Find the maximum likelihood estimator for λ.
Question 7
(5 points) Consider a population governed by the discrete distribution p(x), where p(x) = θ
for x = 1 and p(x) = 1 − θ for x = 0, where θ ∈ [0, 1]. Three independent observations are
{0, 1, 0}. Find the likelihood function
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Question 8
Suppose 5 independent measurements 1, 3, 1, 7, 3 are taken from an exponentially distributed
population f(x) = λe−λx
, where x > 0
(a) (5 points) Find the maximum likelihood estimate for λ.
(b) (4 points) Find the maximum likelihood estimate for the probability that the next independent
measurement taken from the population is greater than 3.