Part 1

Bruce’s Juices Pty Ltd (BJPL) buys all of its oranges from growers in the Milderina region. The amount of juice squeezed from an orange in this region is very well modelled by the normal distribution with mean 47.8 millilitres (ml) and standard deviation 5.4 ml.

(a) (sketch diagram) What proportion of the region’s oranges contains

(i)less than 46.5 ml of juice?

(ii)more than 48.5 ml of juice?

(iii)between 46.5 ml and 48.5 ml of juice?

(b)In a shipment of 100,000 oranges, how many are expected to contain between 46.5 ml and 48.5 ml of juice?

(c)What can be said about the juice content of the least juicy 20% of oranges?

Part 2

The quality control department at Bruce’s Juices tests each incoming shipment. A shipment consists of many hundreds of thousands of oranges.

First, 100 oranges are randomly sampled from a shipment. These oranges are squeezed and the mean amount of juice per orange is calculated. If the sample mean is below 46.5 ml, the shipment is rejected.

(d)

(i)Calculate the probability that a shipment will be rejected.

(ii)For your calculation in (d)(i), how important is the assumption that “the amount of juice squeezed from an orange in this region is very well modelled by the normal distribution”?

Part 3

Bruce’s Juices Pty Ltd has recently been offered fruit by orange growers in the Riverdura region. These growers claim that their oranges are juicier than those of the Milderina region. Assume that juice content of oranges in the Riverdura region is also normally distributed.

(f)A random sample of 250 Riverdura oranges had a mean juice content of 48.2 ml, with standard deviation 4.9 ml. Test the Riverdura growers’ claim at the 1% level of significance.

Thanks in Advance

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