Question

A process used in filling bottles with soft drink results in net weights that are normally distributed, with a mean of 2 liters and a standard deviation of 0.05 liter. Bottles filled to less than 95% of the listed net weight can make the manufacturer subject to penalty by the state office of consumer affairs; bottles filled above 2.10 liters may cause excess spillage upon opening.

What proportion of the bottles will contain

Between 1.90 and 2.0 liters?

At least 2.10 liters?

Ninety-nine percent of the bottles would be expected to contain at least how much soft drink?

The central 40% of bottles will lye between what two liters?

The machine is considered a bargain if it is unlikely to require major repair before the sixth year.

Find the probability that a major repair occurs after 6 years.

Find the probability that a major repair occurs in the first year.

Does the machine seem to be a bargain?

Find the median time before a major repair.

Note: For (a), (b) and (d), don’t forget your concluding statements! For (c), you must give an explanation.

Answer 1

Here it is given that a process used in filling bottles with soft drink results in net weights that are normally distributed, with a mean of 2 liters and a standard deviation of 0.05 liter.

Now we need to find

As x is normally distributed, we can convert x to z

Hence we find that 47.72% of bottles will contain between 1.90 and 2.0 liters

Here we need to find

Again converting x to z we get

Which means 2.28% will contain atlease 2.10 liters

Here we need to find x such that

So first we will use z table to find z such that

Using z table we get

Which mean z=-2.327, for which

Now we will find x using formula of z,

Hence

Hence for Ninety-nine percent of the bottles will contain at least 1.88 liters much soft drink

Here we need to find x1 and x2 such that

Using z table we find that

So

Hence

And similarly

So,

Hence the central 40% of bottles will lie between 1.9738 and 2.0262 liters