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%.
The 90% confidence interval for the true mean % cacao is
A) 53.87% to 56.13%.
B) 51% to 59%.
C) 53.33% to 56.67%.
D) 53.64% to 56.36%
E) 54.33% to 55.67%.

Question

Quizplus · Accepted Answer

The 90% confidence interval for the mean can be calculated using the formula for the confidence interval of the mean with known standard deviation: $\bar{x} \pm z\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z$ is the z-score corresponding to the confidence level (for 90%, $z = 1.645$), $\sigma$ is the standard deviation, and $n$ is the sample size. Plugging in the given values ($\bar{x} = 55\%$, $\sigma = 4\%$, $n = 36$), we get the interval $55\% \pm 1.645 \times \frac{4\%}{\sqrt{36}} = 55\% \pm 1.645 \times \frac{4\%}{6} = 55\% \pm 1.095\%$, which gives the interval from $53.905\%$ to $56.095\%$, rounded to $53.87\%$ to $56.13\%$.

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