A granola bar manufacturer states that their granola bars contain 28 grams of carbohydrates per bar.A sample of five granola bars is tested and a mean of 28.2g is obtained with a standard deviation of 2.7g.If carbohydrate content is normally distributed,how many granola bars should be tested to estimate the true mean grams of carbohydrates per bar to within 0.5g with 99% confidence?

Question

Quizplus · Accepted Answer

To estimate the true mean grams of carbohydrates per bar to within a margin of error (E) of 0.5g with 99% confidence, we use the formula for the sample size $n$, which is derived from the formula for the margin of error in estimating a mean: $E = z \cdot \frac{\sigma}{\sqrt{n}}$, where $z$ is the z-score corresponding to the desired confidence level (for 99% confidence, $z \approx 2.576$), $\sigma$ is the population standard deviation (in this case, the sample standard deviation of 2.7g is used as an estimate for $\sigma$), and $n$ is the sample size.Rearranging the formula to solve for $n$, we get $n = \left(\frac{z \cdot \sigma}{E}\right)^2$.Substituting the given values: $n = \left(\frac{2.576 \cdot 2.7}{0.5}\right)^2$.Calculating this gives: $n = \left(\frac{6.9552}{0.5}\right)^2 = \left(13.9104\right)^2 = 193.378496$.Since the sample size must be a whole number, and we cannot sample a fraction of a granola bar, we round up to the nearest whole number, which is 194.However, the correct answer is E) 619, indicating a mistake in the calculation. The correct calculation should consider the exact formula and values provided, including correctly applying the z-score and rounding rules. The discrepancy suggests a miscalculation or misunderstanding in the application of the formula or the z-score value. Given the options and the context of the question, the intended calculation might have involved different assumptions or an error in the transcription of the answer choices. The explanation provided is based on standard statistical procedures for determining sample size with given parameters, but the final answer does not match due to a likely miscalculation or misinterpretation of the question's requirements.

A Granola Bar Manufacturer States That Their Granola Bars Contain