Question

A bottled water distributor wants to estimate the amount of water contained in 1​-gallon bottles purchased from a nationally known water bottling company. The water bottling​ company's specifications state that the standard deviation of the amount of water is equal to 0.02 gallon. A random sample of 50 bottles is​ selected, and the sample mean amount of water per 1​-gallon bottle is 0.995 gallon. Complete parts​ (a) through​ (d).

a. Construct a 95​% confidence interval estimate for the population mean amount of water included in a​ 1-gallon bottle

? ≤ μ ≤ ?

​(Round to five decimal places as​ needed.)

b. On the basis of these​ results, do you think that the distributor has a right to complain to the water bottling​ company? Why?

(Yes or No), because a​ 1-gallon bottle containing exactly​ 1-gallon of water lies (outside or within) the 95% confidence interval.

c. Must you assume that the population amount of water per bottle is normally distributed​ here? Explain. (Choose the answer below)

A. ​Yes, because the Central Limit Theorem almost always ensures that overbarX is normally distributed when n is large. In this​case, the value of n is small.

B. ​No, because the Central Limit Theorem almost always ensures that overbarX is normally distributed when n is large. In this​case, the value of n is large

C. ​No, because the Central Limit Theorem almost always ensures that overbarX is normally distributed when n is small. In this​case, the value of n is small.

D. ​Yes, since nothing is known about the distribution of the​population, it must be assumed that the population is normally distributed.

d. Construct a 90​% confidence interval estimate. How does this change your answer to part​ (b)?

? ≤ μ ≤ ?

​(Round to five decimal places as​ needed.)

How does this change your answer to part​ (b)?

A​ 1-gallon bottle containing exactly​ 1-gallon of water lies (outside or within) the 90​% confidence interval. The distributor (still has or now has or now does not have or still does not have) a right to complain to the bottling company.

Answer 1

a)

sample mean 'x̄=	0.995
sample size    n=	50.00
std deviation σ=	0.02
std error ='σx=σ/√n=	0.0028

for 95 % CI value of z=		1.96
margin of error E=z*std error =		0.006
lower bound=sample mean-E=		0.98946
Upper bound=sample mean+E=		1.00054

95​% confidence interval =(0.98946 , 1.00054)

b)

No because a​ 1-gallon bottle containing exactly​ 1-gallon of water lies within the 95% confidence interval.

c_) B. ​No, because the Central Limit Theorem almost always ensures that overbarX is normally distributed when n is large. In this​case, the value of n is large

d_)

for 90 % CI value of z=		1.64
margin of error E=z*std error =		0.005
lower bound=sample mean-E=		0.99035
Upper bound=sample mean+E=		0.99965

90% confidence interval =(0.99035 , 0.99965)

A​ 1-gallon bottle containing exactly​ 1-gallon of water lies outside the 90​% confidence interval. The distributor now has a right to complain to the bottling company.