# 求助幾題統計學的問題(20點)

1. A set of final examination grades in an introductory statistics course is normally distributed with a mean of 73 and a standard deviation of 8.

a. What is the probability of getting a grade of 91 or less on this exam?

b. What is the probability that a student scored between 65 and 89?

c. The probability is 5% that a student taking the test scores higher than what grade?

d. If the professor grades on a curve(gives A’s to the top 10% of the class regardless of the score), are you better off with a grade of 81 on this exam or a grade of 68 on a different exam where the mean is 62 and the standard deviation is 3? Show your answer statistically and explain.

2. At an ocean-side nuclear power plant, seawater is used as part of the cooling system. This system raises the temperature of the water that is discharged back into ocean. The amount that the water temperature is raised has a uniform distribution over the interval from 10 to 25℃.

a. What is the probability that the temperature will increase less than 20℃?

b. What is the probability that the temperature will increase between 20 and 22℃?

c. A temperature increase of more than 18℃ is considered to be potentially dangerous to the environment. What is the probability that at any point of time, the temperature increase is potentially dangerous?

d. What is the mean and standard deviation of the temperature increase?

3. Telephone calls arrive at the information desk of a large computer software company at the rate of 15 per hour.

a. What is the probability that the next call will arrive within 3 minutes (0.05 hour)?

b. What is the probability that the next call will arrive within 15 minutes (0.25 hour)?

### 1 個解答

• 1 0 年前
最佳解答

1. 已知 X ～ N( μ = 73，σ² = 8² )

a.

X - μ　　　91 - 73

P( X 小(等)於 91 ) = P( ------------- < ------------- )

σ　　　　8

= P( Z < 2.25 )

= 0.988

b.

65 -73　　　X - μ　　89 - 73

P( 65 < X < 89 ) = P( --------------- < ------------ < ------------- )

8　　　　　σ　　　　8

= P( -1 < Z < 2 )

= P( Z < 2 ) - P( Z < -1 )

= 0.977 - 0.159

= 0.819

c.

X - μ　　　k - 73

P( X > k ) = P( ------------- > ------------- )

σ　　　　8

= P( Z > 1.645 )

= 0.05

k - 73

所以 ------------ = 1.645 ， k = 73 + 1.645 × 8 = 86.160

8

d.

this exam：X ～ N( μx = 73，σx² = 8² )

different exam：Y ～ N( μy = 62，σy² = 3² )

X - μx　　81 - 73

P( X < 81 ) = P( ------------- < ------------- ) = P( Zx < 1 ) ------- i

σx　　　8

Y - μy　　68 - 62

P( Y < 68 ) = P( ------------- < ------------- ) = P( Zy < 2 ) ------- ii

σy　　　3

由 i 、 ii 得知，本次考試成績（X）並沒有比另一場考試成績（Y）優。

2. 已知 X ～ U( 10，25 )

a.

1

P( X < 20 ) = ( 20 - 10 ) × ----------------- = 0.667

25 - 10

b.

1

P( 20 < X < 22 ) = ( 22 - 20 ) × ------------------- = 0.133

25 - 10

c.

1

P( X > 18 ) = ( 25 - 18 ) × ------------------ = 0.467

25 - 10

d.

10 + 25

E(X) = ----------------- = 17.5

2

( 25 - 10 )

V(X) = ------------------ = 18.75

12

3.已知 X～ Poi（λ = 15次 / hr ）

e^λ × λ^x

所以 f( X ) = --------------------，其中 x = 0, 1, 2, …。

x !

a.

e^0.75 × 0.75^1

P( X = 1 ) = ------------------------- = 0.354

1 !

註：λ = 15次 / hr = 15次 / 60 min = 0.75次 / 3 min

b.

e^3.75 × 3.75^1

P( X = 1 ) = ------------------------- = 0.88

1 !

註：λ = 15次 / hr = 15次 / 60 min = 3.75次 / 15 min

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