Question

The fill amount of bottles of a soft drink is normally​ distributed, with a mean of

2.0

liters

and a standard deviation of

0.05

liter. Suppose you select a random sample of

25

bottles.

a. What is the probability that the sample mean will be between

1.99

and

2.0

liters​?

b. What is the probability that the sample mean will be below

1.98

liters?

c. What is the probability that the sample mean will be greater than

2.01

​liters?

d. The probability is

90%

that the sample mean amount of soft drink will be at least how​ much?

e. The probability is

90​%

that the sample mean amount of soft drink will be between which two values​ (symmetrically distributed around the​mean)?

Answer 1

Given data

mean μ =2.0

standard deviation σ = 0.05

number of samples n = 25

a. What is the probability that the sample mean will be between 1.99 and 2.0

for x=1.99 ⇒ Z = (x-μ)/(σ/√n) = (1.99-2.0)/(0.05/√25) = -1

forx= 2 ⇒ Z = (x-μ)/(σ/√n) = (2-2)/(0.05/√25) = 0

P(1.99 < x̅ < 2) = P(-1 < z < 0) = P(Z<=0)-P(Z<=-1) = 0.5-0.15866 = 0.34134

P(1.99 < x̅ < 2.0) = 0.34134

b. What is the probability that the sample mean will be below x =1.98

z = (x-μ)/(σ/√n) = (1.98-2.0)/(0.05/√25) = -2

P ( X < 1.98 ) = P ( Z < -2 ) = 0.0228

P ( X < 1.98 ) = 0.0228

c.What is the probability that the sample mean will be greater than x = 2.01

z = (x-μ)/(σ/√n) = (2.01 - 2.0)/(0.05/√25) = 1

P ( X > 2.01) = 1 - P(z<1) = 1 - 0.8413 = 0.1587

P ( X > 2.01) =  0.1587

d.The probability is 90% that the sample mean amount of soft drink will be at least how mucc

P( z ≥ ? ) = 0.90

P( z ≥ ? ) = 1 - P( z ≤? ) = 1 - 0.90 = 0.1

z_{0.1 = -1.28}

z = (x-μ)/(σ/√n) ⇒ -1.28 = (x-2)/(0.05/√25)

x-2 = -0.0128

x = 2-0.0128

x = 1.98

e. The probability is 90​% that the sample mean amount of soft drink will be between

which two values​ (symmetrically distributed around the​mean)?

P ( a < Z < b ) = 0.90

Area above the mean = 0.5 + ( 0.90 / 2 ) = 0.95

i.e z_{0.95 = ±1.65}

z = (x-μ)/(σ/√n) ⇒ -1.65 = (x-2)/(0.05/√25)

x-2 = -0.0165

x = 2 - 0.0165

x = 1.98

z = (x-μ)/(σ/√n) ⇒ 1.65 = (x-2)/(0.05/√25)

x-2 = 0.0165

x = 2 + 0.0165

x = 2.01

P ( 1.98 < X < 2.01 ) = 90%