Grandma Gertrude's Chocolates, a family owned business, has an opportunity to
supply its product for distribution through a large coffee house chain. However, the
coffee house chain has certain specifications regarding cacao content as it wishes to
advertise the health benefits (antioxidants) of the chocolate products it sells. In order to
determine the mean % cacao in its dark chocolate products, quality inspectors sample 36
pieces. They find a sample mean of 55% with a standard deviation of 4%.
A. Describe the sampling distribution for the sample mean.
b. What is the standard error?
c. What is the margin of error for 90% confidence?
d. What is the margin of error for 95% confidence?
e. Based on the sample results, find the 90% confidence interval and interpret.
f. Based on the sample results, find the 95% confidence interval and interpret.
g. For a more accurate determination of the mean weight, the quality control inspectors
wish to estimate it within 1% with 95% confidence. How many pieces of dark chocolate
should they sample?

Question

Quizplus · Accepted Answer

The Answer is a. Describe the sampling distribution for the sample mean.
The sampling distribution for the sample mean can be modeled using the

t

-distribution with 35 degrees of freedom.
b. What is the standard error?

S E ( \bar { y } ) = \frac { s } { \sqrt { n } } = \frac { 4 } { \sqrt { 36 } } = 0.67 \%

c. What is the margin of error for

90 \%

confidence?

\dot { t } _ { n - 1 } ^ { * } imes S E ( \bar { y } ) = 1.690 imes 0.67 = 1.13 \%

d. What is the margin of error for

95 \%

confidence?

\dot { t } _ { n - 1 } ^ { * } imes S E ( \bar { y } ) = 2.030 imes 0.67 = 1.36 \%

e. Based on the sample results, find the

90 \%

confidence interval and interpret.

\bar { y } \pm t _ { n - 1 } ^ { * } imes S E ( \bar { y } ) = 55 \pm 1.13

The

90 \%

confidence interval for true

\%

cacao is

53.87 \%

to

56.13 \%

. We are

90 \%

confident that the mean percentage of cacao in Grandma Gertrude's dark chocolate is between

53.87 \%

and

56.13 \%

.
f. Based on the sample results, find the

95 \%

confidence interval and interpret.

\bar { y } \pm t _ { n - 1 } ^ { * } imes S E ( \bar { y } ) = 55 \pm 1.36

The

95 \%

confidence interval for true

\%

cacao is

53.64 \%

to

56.36 \%

. We are

95 \%

confident that the mean percentage of cacao in Grandma Gertrude's dark chocolate is between

53.64 \%

and

56.36 \%

.
g. For a more accurate determination of the mean weight, the quality control inspectors wish to estimate it within

1 \%

with

95 \%

confidence. How many pieces of dark chocolate should they sample?

n = \frac { ( 1.96 ) ^ { 2 } ( 4 ) ^ { 2 } } { 1 ^ { 2 } }

n = 61.46

or 62 pieces of chocolate

Grandma Gertrude's Chocolates, a Family Owned Business, Has an Opportunity