STAT410- Homework #11 Solved

 

10- 11.             A jackleg Discord user uses Discord mainly just to "collect" the homework

answers (that is, not your average Discord user, since the majority of Discord users are, of course, on Discord to learn, allegedly). Suppose that the time the jackleg Discord user needs to spend on Discord to "collect" the homework answers varies from assignment to assignment according to a normal distribution with the overall standard deviation G = 12 minutes. We wish to use (a random sample of) n = 9 homework assignments to test . 

10.               H :                 60 minutes versus HI : > 60 minutes.

a)                  If the sample mean is x = 65.4 minutes, find the p-value of the test.

b)                 Find the Rejection Region for the test at u = 0.05. That is, for which values of the sample mean X should we reject HO, if a 5% level of significance is used?

c)                  Find the power of the test if a 5% level of significance is used, and the true value of the average amount of time the jackleg Discord user needs to spend on Discord to "collect" the homework answers is (i) = 63.1 minutes, (ii) = 67.9 minutes.

11.                     Ho: PI = 60 minutes versus HI : VI 60 minutes.

a)                  If the sample mean is x = 65.4 minutes, find the p-value of the test.

b)                 Find the Rejection Region for the test at u 0.05. That is, for which values of the sample mean X should we reject HO, if a 5% level of significance is used?

c)                  Find the power of the test if a 5% level of significance is used, and the true value of the average amount of time the jackleg Discord user needs to spend on Discord to "collect" the homework answers is (i) = 63.2 minutes, (ii) = 54.2 minutes.

12- 13.        Recall Homework #07 :

6.         After a homework assignment is posted on Compass2g, the weak-minded lazy Chegg enthusiasts flock to Chegg according to a Poisson process with the average rate of I .5 weak-minded lazy Chegg enthusiasts per minute. ( The pudding-brain lazy CourseHero worshipers show similar behavior. )

Let  the number of weak-minded lazy Chegg enthusiasts who show up in t minutes,  the time when the kth weak-minded lazy Chegg enthusiast shows up.

Then

Xt has a Poisson (X t) distribution,

1

                                     Tk has a Gamma (U = k, O          ) distribution.

A weak-minded lazy Chegg enthusiast teams up with a pudding-brain lazy CourseHero worshiper in an attempt to show that the average rate of weak- minded lazy Chegg enthusiasts flocking to Chegg after a homework assignment is posted on Compass2g is less than 1.5 weak-minded lazy Chegg enthusiasts per minute. They wish to test

                                                   1.5        versus         HI : < 1.5.

However, they are "a little confused" about how to do it and cannot agree on the plan.

12. The weak-minded lazy Chegg enthusiast wants to consider the number of weak-minded lazy Chegg enthusiasts who show up in 7 minutes, X 7, and then Reject HO if X 7 is too small.

a)                  Help the weak-minded lazy Chegg enthusiast find the best rejection region with the significance level u closest to 0.05.

Hint 1: Reject Ho if x 7 < c.                         Hint2: X 7        Poisson(7X).

Hint 3: Want c such that 0.05 = u = P (Reject Ho I HO is = 1.5).

b)                 Find the power of the rejection region from part (a) if = 1.

Hint:    I Ho is NOT true) = P(X7 

c)                  Consider the rejection region

Reject HO if at most 6 weak-minded lazy Chegg enthusiasts show up in 7 minutes.

That is, Reject HO if X 7 6. Find the significance level u for this rejection region.

Hint:  u = P (Reject HO I HO is true) = P (X 7 S 6 X = 1.5 ).

d)                 Suppose that 4 weak-minded lazy Chegg enthusiasts show up in 7 minutes. Find the p-value of the test.

13.       The pudding-brain lazy Coursel--lero worshiper wants to consider the time when the 4 th weak-minded lazy Chegg enthusiast shows up, T 4, and then Reject HO if T 4 is too large.

a)                  Help the pudding-brain lazy CourseHero worshiper find the best Rejection Region with the significance level u = 0.05.

Hint 1: Reject Ho if T 4 2 c.                                   Hint 2: T 4 Gamma(u = 4, 0 l//x).

Hint 3: Want c such that 0.05 = u = P (Reject HO I HO is true)   I X = 1.5 

Hint 4 (suggestion):           If T has a Gamma(u, 0 l//x) distribution, then

2 T//0 = 2 XT has a 1 2 (20) distribution.

b)                 Consider the rejection region

Reject HO if we wait more than 5 minutes for the 4 th weak-minded lazy Chegg enthusiast to show up.

That is, Reject HO if T 4 > 5. Find the significance level u for this rejection region. Hint l:   1 X = 1.5).

Hint 2 (suggestion): If T has a Gamma(u, 0 I//x) distribution, where 01 is an integer, then FT(t) = P(TSt) = pc x t 20) and = P(xts a- l), where Xt has a Poisson(Xt) distribution.

c)                  Find the power of the rejection region from part (b) if X = l .

Hint:         I Ho is NOT true)   1 X = 

d)                 Suppose that the 4 th weak-minded lazy Chegg enthusiast shows up at the very end of the 6 th minute. Find the p-value of the test.

 
 

1 DONT 

TO DO STATISTICS BUT

IT DOESN'T MATTER BECAUSE 1 DIDNT HAVE DATA.