A fruit juice company claims that the cups of juice dispensed from their vending machines have a mea

Question

A fruit juice company claims that the cups of juice dispensed from their vending machines have a mean volume of $275$ ml. A quality control officer collects a random sample of $40$ cups of juice from various vending machines. The volumes, $x$, in ml, are summarised by $$\sum (x-275) = -48, \quad \sum (x-275)^2 = 539.95$$ Suggest why, in this context, the data is summarised in terms of $(x-275)$ rather than $x$.[1] Find the unbiased estimates of the population mean and variance.[3] Test at the $5$% level of significance whether the claim of the fruit juice company is valid.[4] The company makes some improvement to their vending machines and now decides to take a new random sample of $50$ cups of juice to investigate the situation. The volumes of juice in a cup are now assumed to have a normal distribution with standard deviation $3.5$ ml. The mean volume of juice in this sample is $k$ ml. A test at the $4$% significance level gives insufficient evidence that the population mean volume of juice is less than $275$ ml. Find the range of possible values of $k$.[4]

Accepted Answer

(ii)$\bar{x}=273.8, s^2=12.4$ (iii)p-value = 0.0309 (iv)k>274

2024 MI H1 Q13

2024 MI H1 Q13