A coffee shop owner claims that its signature coffee has a mean caffeine content of $120$ mg per c

Question

A coffee shop owner claims that its signature coffee has a mean caffeine content of $120$ mg per cup. To investigate this claim, a random sample of $60$ cups of coffee is selected and the caffeine content, $x$ mg per cup, is summarised as follows.$$\sum x = 6960, \quad \sum x^2 = 822465$$ Calculate unbiased estimates of the population mean and variance for the caffeine content per cup.[2] Test, at the $5$% level of significance, whether the coffee shop owner’s claim is supported by the data. You should state your hypotheses and define any symbols that you use.[4] The coffee shop also sells premium coffee, where its caffeine content is normally distributed with population variance $200$ mg${}^2$. The coffee shop owner now claims that its premium coffee has a mean caffeine content less than $120$ mg per cup. Another large random sample of $n$ cups of premium coffee is selected and the sample mean caffeine content per cup is found to be $116.6$ mg. A test is carried out at the $2\frac{1}{2}$% level of significance and the result supports the owner’s claim. Find the set of values that $n$ can take.[4]

Accepted Answer

(a)$\bar{x} =116$, $s^2= 256$ (c){$n\in\mathbb{Z}, n \geq 67$}

2025 TMJC P2 Q7

2025 TMJC P2 Q7