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日期：2023-05-04 08:42

Question 1 (40 points)

Let   = ( !, … ,  ") be i.i.d. random variables from the Rayleigh distribution:

( ;  ) = #

$!  %#!/'$!

,   > 0,   > 0

Note that as   → ∞,  1 = 2 !

'" ∑  ( " '

()!

*

→   5 ,

$!

+"

6. Using this result, do the following:

a) Construct an approximate 95% confidence interval for  . Calculate this confidence

interval using the data found in rayleigh.txt.

b) Construct an approximate 95% confidence interval for   =  8 /2 , which

happens to be the expected value of the Rayleigh distribution. Calculate this

confidence interval using the data found in rayleigh.txt.

Now, we’d like to test the following hypothesis at a 1% level of significance using a

maximum likelihood test statistic:

,:   = 1 versus  -:   ≠ 1

c) by defining appropriate acceptance and rejection regions. The data necessary for

this test can be found in rayleigh.txt. What do you conclude?

d) by calculating the p-value. The data necessary for this test can be found in

rayleigh.txt. What do you conclude?

2

Question 2 (60 points)

Online retailers often use banner ads to drive traffic to their websites. Nike is trying to

determine which of two online banner ads (AD1 vs. AD2) is more effective at increasing

online sales. To investigate which is better, AD1 is displayed in Facebook newsfeeds for

IP addresses originating in the Western US and AD2 is displayed in Facebook newsfeeds

for IP addresses originating in the Eastern US. Both ads are run for the month of February,

and on each day the revenue from www.nike.com is recorded for Eastern and Western IP

addresses. This revenue data (in $100,000s) can be found in nike.txt. The question of

interest is to decide which (if either) of the Eastern or Western revenues tends to be larger.

This would indicate which ad campaign is more profitable.

Note that an experiment like this, in which two conditions are compared to decide which

is better, is often referred to in marketing and technology literature as A/B testing.

For the purposes of this assignment, we may assume that revenues associated with AD1

and AD2 are normally distributed with unknown means and unknown standard deviations.

Notation:

? Let  (.~ B .,  .

'E denote the revenue associated with ad   = 1,2 on day   =

1,2, … ,  

? Let  H. =  I

. = !

" ∑  (. "

()! be the ML estimator of  .,   = 1,2

? Let  . = 2 !

"%! ∑ B (. ?  I

.E

' "

()! be an estimator of  .,   = 1,2

(a) Construct an exact 95% confidence interval for  !, the average revenue associated

with AD1.

(b) Construct an exact 95% confidence interval for  ', the average revenue associated

with AD2.

(c) Define  H! ?  H' =  I

! ?  I' to be the estimator of  ! ?  ' , the true difference

between AD1 and AD2 average revenues. Assuming the two populations are

independent, what is the distribution of  H! ?  H'? Use this distribution to define a

pivotal quantity that is a function of  I!,  I',  !,  ',  !,  '. With this pivotal quantity,

construct an exact 95% confidence interval for  ! ?  '.

(d) With the test statistic/pivotal quantity from (c), test the following hypothesis at a

5% level of significance:

,:  ! =  ' versus  -:  ! ≠  '

(e) Using your test statistic from (d), at a 5% level of significance test the following

hypothesis:

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,:  ! ≤  ' versus  -:  ! >  '

(f) Based on the confidence intervals and hypothesis tests in calculated in (a)-(e),

comment on which of the two ads seems to be associated with higher revenue. Nike

plans to run the better of the two ads in all of North America. Which ad should Nike

choose? Why?