Answer to Question #209198 in Statistics and Probability for Jayz

Question

A particular brand of coffee contains an average of 112 mg of caffeine per cup with a standard deviation of 20 mg. A barista wants to investigate the same to estimate the true population mean caffeine content correct to within 5 mg adopting 95% confidence. How many cups of the same brand of coffee does he need for a sample?

Answer 1

The critical value for $\alpha=0.05$ is $z_c=z_{1-\alpha/2}=1.96$

The corresponding confidence interval is computed as shown below:

$$CI=(x-z_c\times \dfrac{\sigma}{\sqrt{n}}, x+z_c\times \dfrac{\sigma}{\sqrt{n}})$$

$$SE=z_c\times \dfrac{\sigma}{\sqrt{n}}=>n=( \dfrac{z_c\sigma}{SE})^2$$

Substitute

$$n=( \dfrac{1.96(20)}{5})^2=61$$

He needs 61 cups of the same brand of coffee for a sample.