Bottles of apple juice are said to contain $600$ ml. A random sample of $60$ bottles is taken an

Question

Bottles of apple juice are said to contain $600$ ml. A random sample of $60$ bottles is taken and the volumes (in ml) of apple juice, $x$, in the bottles are measured and are summarised by

\[\sum(x-600)=180, \quad \sum(x-600)^2=6500.\]

Find unbiased estimates for the population mean and variance.[3]
Test at the $5$% significance level, whether there is evidence that the population mean volume of apple juice is understated.[4]

Bottles of orange juice are said to contain $300$ ml. It is given that the volume of orange juice is normally distributed and the population variance is $100$ $\mathrm{ml}^2$. A random sample of $15$ bottles is taken and the mean volume in this sample is $\bar{y}$ ml. A test at the $5$% significance level indicates that there is sufficient evidence to conclude that the population mean volume differs from $300$ ml.

Find the range of values of $\bar{y}$.[3]

Accepted Answer

(i) $\bar x=603$ ml, $s^2=\frac{5960}{59}\approx101.017$ ml$^2$. (ii) $p\approx0.0104$; reject $H_0$, sufficient evidence that the mean is understated. (iii) $\bar y300+\Phi^{-1}(0.975)\sqrt{\frac{100}{15}}$.

2024 TMJC H1 Q11

2024 TMJC H1 Q11

Solution:

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