Question

The consumer magazine also claims that the cinnamon rolls at STARBUCKS do not equal 13 ounces. A random sample of 25 customers purchasing cinnamon rolls yields the following results:- the sample mean equals 13.87 ounces - it is known from previous studies that the population standard deviation equals 0.25 ounces.

a. Set up a 95% confidence interval for the true mean?

b. What sample size is required if you want to be 90% sure that the sample mean will be within 0.2 ounces of the true mean?

c. Test the hypothesis that the true population mean is not equal to 13 ounces. Set the type one error equal to 5%.

Answer 1

a)

95% Confidence Interval :-
X̅ ± Z( α /2) σ / √ ( n )
Z(α/2) = Z (0.05 /2) = 1.96
13.87 ± 1.96 * 0.25/√(25)
Lower Limit = 13.87 - 1.96 * 0.25/√(25)
Lower Limit = 13.772
Upper Limit = 13.87 + 1.96 * 0.25/√(25)
Upper Limit = 13.968
95% Confidence interval is ( 13.77 , 13.97 )

b)

Sample size = (Z/2 * / E)2

= ( 1.6449 * 0.25 / 0.2)2

= 4.23

Sample size = 5 (Rounded up to nearest integer)

c)

H0: = 13

Ha: 13

Test Statistic :-
Z = ( X̅ - µ ) / ( σ / √(n))
Z = ( 13.87 - 13 ) / ( 0.25 / √( 25 ))
Z = 17.4
Test Criteria :-
Reject null hypothesis if | Z | > Z( α/2 )
Critical value Z(α/2) = Z( 0.05 /2 ) = -1.96 , 1.96
| Z | > Z( α/2 ) = 17.4 > 1.96
Result :- Reject null hypothesis