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 is 0.02 gallon. You select a random sample of 45 ​bottles, and the mean amount of water per​ 1-gallon bottle is 0.994 gallon. a. Is there evidence that the mean amount is different from 11.0 ​gallon? (Use α=0.01​.) Let μ be the population mean. Determine the null​ hypothesis, H0​, and the alternative​ hypothesis,H1.

H0​:μ = 1

H1​:μ≠1

Compute the​ p-value and interpret its meaning. What is the​ p-value? _____

Answer 1

Solution:

Given:

Population Standard Deviation =

Sample size = n = 45

Sample mean =

Population Mean =

We have to test if there is evidence that the mean amount is different from 1.0 ​gallon.

Thus this is two tailed test.

Thus H0 and H1 are:

Compute p-value and interpret its meaning.

To find p-value , we need to find z test statistic.

Thus p-value is:

p-value = 2 X P( Z < z test statistic value)

( We multiply by 2 , since this is two tailed test and we take Z < z value , since z is negative, if z is positive we should take Z > z )

Thus we get:

p-value = 2 X P( Z < -2.01)

Look in z table for z = -2.0 and 0.01 and find area.

P( Z < -2.01) =0.0222

Thus

p-value = 2 X P( Z < -2.01)

p-value = 2 X 0.0222

p-value = 0.0444

Interpretation:

p-value = 0.0444 > 0.01 level of significance , we fail to reject null hypothesis at 0.01 significance level.

Thus there is not sufficient evidence that the mean amount is different from 1.0 ​gallon.