Question

A bottled water distributor wants to determine whether the mean amount of water contained in 1-gallon bottles purchased from a nationally known water bottling company is actually 1 gallon. You know from the water bottling company specifications that the standard deviation of the amount of water per bottle is 0.03 gallon. You select a random sample of 100 bottles, the mean amount of water per 1-gallon bottle is 0.994 gallon.

a. Is there evidence that the mean amount is different from 1.0 gallon? (Use ? = 0.01)

b. Compute the p-value and interpret its meaning.

c. Construct a 95% confidence interval estimate of the population mean amount of water per bottle.

d. Compare the results of (a) and (c). What conclusions do you reach?

Answer 1

x̅ = 0.994, σ = 0.03, n = 100      

a) Null and Alternative hypothesis:

Ho : µ = 1  

H1 : µ ≠ 1  

Critical value :   

Two tailed critical value, z crit = ABS(NORM.S.INV(0.01/2 )) = ± 2.576

Test statistic:   

z = (x̅- µ)/(σ/√n) = -2.0

As |z| = 2 < 2.576, we fail to reject the null hypothesis.

There is not evidence that the mean amount is different from 1.0 gallon at 0.01 significance level.

b) p-value = 2*(1-NORM.S.DIST(ABS(-2.0, 1) = 0.0455

c) 95% Confidence interval :

At α = 0.05, two tailed critical value, z crit = NORM.S.INV(0.05/ 2) = 1.960

Lower Bound = x̅ - z-crit*σ/√n = 0.9881

Upper Bound = x̅ + z-crit*σ/√n = 0.9999

d) The result of a) and c) are not same.

In a) we fail to reject the null hypothesis.

In c) we reject the null hypothesis that the population mean is equal to 1 gallon.