Statistical Analysis (estimation, distribution, etc.)

Question 4

Let x₁, x₂, …xₓ be a random sample of n i.i.d. observations
following the Poisson distribution with parameter (mean) ϴ such that E(Xₐ) = ϴ.  Consider three estimators of ϴ given as:

ϴ₁ = X,     ϴ₂ = X₁,   ϴ₃ = (x₁ + 2x₂)/3

i.) 
Show that all the three estimators are unbiased

ii.) 
Calculate which of three has the least Variance

Question 5

A psychology researcher wants to find out if exercising
before taking a quiz affects a student’s performance.  To test this he randomly assigns students to
either exercise for 10 minutes before taking a short quiz, or to take the quiz
without exercising first.  37 students
exercise first and averaged 84% on the quiz with a standard deviation of 7%
while 32 students skip exercising and score 81% with a standard deviation of
6%.  Neither sample was substantially
skewed.

a.) 
Write down the null and the alternative
hypothesis in the test.

b.) 
Draw a picture of the samlpling distribution
assuming that the null hypothesis is true, shading in the regions(s) that would
result in a type 1 error (falsely rejecting the null hypothesis assuming that
it’s true of 0.05

c.) 
Test the hypothesis at 95% confidence.  NOTE: 
Instead of assuming that the variances are equal, first test for the
equality of variances.  Once you conclude
regarding the equality of variances, test for the equality of mean performance.

d.) 
What is the P-value of the test?

Question 6

a.) 
For the IND ENG 716 group project, members of
group 5 needed to determine if the percentage titanium content std. dev. In an aerospace
casting is within the acceptable specifications of SD = .30.  To do this, they tested the percentage
content of titanium in 51 casting specimen and the sample std. dev. was 0.37.  Construct the 95% confidence interval of the
standard deviation and comment on whether the company’s castings are within the
allowable specifications

b.) 
Next the group needed to study if the two ally
impurity level detection tests that the company uses perform the same.  To do this, they tested eight specimens on
both apparatus and recorded the following impurity levels:

Is there evidence at 90% confidence that
the tests differ in mean impurity levels? 
Hint:  be careful to decide if
these are homogenous samples or not, hence use the appropriate test.  What is the P-value of your test?