Question

Bottled water distributors want to determine whether the contents of the 1 gallon bottle are actually 1 gallon. It is known from the specifications of water bottling companies that the standard deviation of water content per bottle is 0.02 gallons. If a random sample of 50 bottles is chosen, and the average sample of water content per 1 gallon bottle is 0.995 gallons.
a. Is there any evidence that the average water content in bottles is not the same as 1.0 gallons? Use α = 0.01.)
b. Calculate the p-value and interpret the meaning.

P.S No handwritten answer would be very appreciated

Answer 1

a)

H0: = 1

Ha: 1

Test Statistic :-
Z = ( X̅ - µ ) / ( σ / √(n))
Z = ( 0.995 - 1 ) / ( 0.02 / √( 50 ))
Z = -1.77

Decision rule :-

Reject null hypothesis if | Z | > Z( α/2 )
Critical value Z(α/2) = Z( 0.01 /2 ) = 2.576

| Z | > Z( α/2 ) = 1.77 < 2.576
Result :- Fail to reject null hypothesis

Conclusion - We do not have sufficient evidence to support the claim that the average water

content in bottles is not the same as 1.0 gallons

b)

For two tailed test,

P value = 2 * P(Z < z )

= 2 * P ( Z < -1.77 )

= 2 * 0.0384 (From Z table)

= 0.0768

Assuming nul hypothesis is true, average difference or more 7.68% of the times dues to sampling error.