STATISTICS ASSIGNMENT ATTACHED

1. (30pts) Problem 8.2.6, page 332.

a. Run a Hartley test, use 05.0 . b. If you accept 𝐻𝑜 in a, run an ANOVA, use 05.0 . c. If you reject 𝐻𝑜 in b, do the Tukey test, use 05.0 . 2. (5pts) Suppose the weights W (kg) of a male population is normally distributed. If we know 𝑃(𝑊 ≤ 70) = 0.65, and 𝑃(𝑊 ≤ 50) = 0.35. Now from this population, 15 males are randomly selected, what is the probability that at least 3 of them weigh more than 65kg? 3. (5pts) If after a Tukey test (𝑘 populations), and you reach the statistical conclusion that 𝜇𝑖 ≠ 𝜇𝑗 , use what you have learned from

this class to verify that the confidence interval for 𝜇𝑖 − 𝜇𝑗 is the following

(𝑥𝑖 − 𝑥𝑗 ) − # ∙ 𝑀𝑆𝑊 ∙ √ 1

𝑛𝑖 +

1

𝑛𝑗 < 𝜇𝑖 − 𝜇𝑗 < (𝑥𝑖 − 𝑥𝑗 ) + # ∙ 𝑀𝑆𝑊 ∙ √

1

𝑛𝑖 +

1

𝑛𝑗

where # is come from t-distribution. What is the degree of freedom of this t-distribution? 4. (5pts) When testing 𝐻𝑜 : 𝜇1 = 𝜇2 ↔ 𝐻𝑎 : 𝜇1 ≠ 𝜇2, in case 𝜎1, 𝜎2 are unknown but 𝜎1 = 𝜎2, we use the following formula

(𝑥1−𝑥2)−(𝜇1−𝜇2)

√ (𝑛1−1)𝑠1

2+(𝑛2−1)𝑠2 2

𝑛1+𝑛2−2 √

1

𝑛1 +

1

𝑛2

= 𝑡𝑛1+𝑛2−2.

In case 𝜎1, 𝜎2 are unknown but 𝜎1 = 3𝜎2, find the right formula.

5. (5pts) Let 1

x ,…, 16

x be an independent sample from a normal distribution ,(N 5). For the following test problem:

5.6:5.6:   ao

HH , let the rejection region be }|5.6{| cxW  (Note, in the lecture, I often call the

rejection region as “the wired region”). Find c to make the significant level, namely , of this test be 0.05. 6. (25pts) There are seven types of artificial man-made fibers, 4 pieces were taken from each type to test their strengths. The summery of the date is given as the following:

Types 1 Type 2 Type 3 Type 4 Type 5 Type 6 Type 7 𝑥1 = 6.3 𝑥2 = 6.2 𝑥3 = 6.7 𝑥4 = 6.8 𝑥5 = 6.5 𝑥6 = 7.0 𝑥7 = 7.1

𝑠1 = .81 𝑠2 = 0.92 𝑠3 = 1.22 𝑠4 = .74 𝑠5 = .88 𝑠6 = .58 𝑠7 = 1.05

a. With 𝛼 = 5%, run Hartley test to test 𝐻𝑜 : 𝜎1 = 𝜎2 = ⋯ = 𝜎7 b. If you accept 𝐻𝑜 in a, with 𝛼 = 5%, run ANOVA to test 𝐻𝑜 : 𝜇1 = 𝜇2 = ⋯ = 𝜇7 (assume all dates are from normal

distributions) c. If you accept 𝐻𝑜 in b (that means all date from the same population), please find a 95% confidence interval for 𝜇 (𝜇 = 𝜇1 =

𝜇2 = ⋯ = 𝜇7); if you reject 𝐻𝑜, please run the follow up Tukey text with 𝛼 = 5%. 7. (5pts) Suppose that two drugs are under study. The researcher is willing to assume that response times, following administration of two drugs, are normally distributed with equal variance of 68. As part of the evaluation of two drugs, drug A was administered to

16 subjects and drug B to administered to 19 subjects. If the difference of sample mean, namely BA

xx  , is 3.6, can you find a

number k such that %90)(  kP BA

 ? If so, find this number k .

8. (5pts) Let 1

x ,…, n

x be an independent sample from a normal distribution ,(N ) . Let 2

1

2 )(

1 xx

n S

n

i

in  



(Note,

this is different from 2

1

2 )(

1

1 xx

n s

n

i

in 

  



, but related!). Find the smallest n such that

.8.0} 2

1 |{|

222  

n SP

9. (5pts) A cell has only two forms, from A and form B. On any given day to the next, the cell either stay in the same from with probability 0.65 or change to other from with probability 0.35. If the cell is in A form today, in a month (30 days), it is either in A form or in B form, what is the probability that the cell is in A form in a month? 10. (10pts) Problem 12.4.6, page 629.