Quality Assurance & Estimation (Edexcel GCSE Statistics: Higher)

A company fills bottles with spring water.

The bottles of spring water have a target volume of 500 ml.

Samples of the bottles of water are taken from the production line at regular intervals and the mean volume of water in the bottles in each sample is found.

The sample means should be normally distributed with a mean of 500 ml and a standard deviation of 6 ml.

Find the lower action limit of the sample means for the bottles of water.

The manager in charge of monitoring the filling of the water bottles wants to set the lower action limit further below the target volume of 500 ml.

What effect would you expect this to have on the number of times the production process may need to be stopped?

The company also fills cans of fizzy drink.

Here are the quality control charts for the sample means and the sample ranges of the volume of fizzy drink in the cans.

A sample of cans of fizzy drink is taken.

The mean of the sample is 346 ml.
The range of the sample is 12 ml.

Determine what action, if any, should be taken.
Justify your answer by referring to both the sample mean and the sample range.

An environmentalist is collecting information about a population of frogs living in a pond.

He uses secondary data from a previous investigation completed by someone else.

In the original investigation, a sample of frogs were caught and tagged before being released back into the pond.

Later, another random sample of frogs from the same pond were caught and the number that had a tag was recorded.

The data from the investigation is summarised in the table below.

Work out an estimate for the total number of frogs living in the pond.

Discuss the reliability of using secondary data to work out an estimate for the total number of frogs in the pond.

It is suggested that the time between the two samples being collected was quite short.

If this had happened, describe the effect it would have on the estimate worked out in part (a).

A food company makes packets that are filled with jelly sweets.

For quality control, a sample of 4 packets of jelly sweets is taken at regular intervals and the mean weight of the packets in the sample is calculated.

When the packaging machine is working properly, the mean weight, in grams, in a sample of 4 packets of jelly sweets is modelled by the normal distribution .

A quality control chart for the sample means is drawn.

Five sample means have been plotted.

Complete the control chart by drawing the lower warning line, the upper warning line and the upper action line.
Label your lines.

The sample mean of sample 6 is 165 g.

Determine any actions that need to be taken.

Felix says that the standard deviation of the weights of the packets of jelly sweets should be more than 4 g.

Is Felix correct?

Ankunda wants to estimate the number of deer in a forest.

He catches a sample of 28 deer. Each deer is tagged and then released back into the forest.

Two weeks later, Ankunda catches a sample of 35 deer in the forest.

He uses the Petersen capture recapture formula to estimate that there are 140 deer in the park.

Work out how many of the 35 deer in Ankunda's sample had tags.

Comment on the reliability of Ankunda's estimate by considering the assumptions he needed to make in order to use the Petersen capture recapture formula.

A company manufactures screws for a shelving unit.

For quality assurance, the company takes samples of 4 randomly selected screws at regular intervals to check the mean length of the 4 screws.

When the machine is working properly, the mean length of a screw, in millimetres, in a sample of 4 screws is modelled by the normal distribution .

The company uses quality control charts to monitor the manufacturing process.

Complete the quality control chart below for the company to use.

Chris is in charge of quality control.

He takes samples of 4 screws at regular intervals throughout the day. In 30 samples, Chris finds that there is one sample that exceeds the warning limit but lies within the action limit.

He decides to shut down the machine and investigate the problem further.

Explain whether or not Chris' actions are appropriate with respect to quality control.

Hannah claims that the length of individual screws should be modelled by the same normal distribution as the mean length per screw in a sample of 4 screws.

Explain whether or not Hannah’s claim is justified.
You must give reasons for your answer.

A food manufacturer packages tins of beans of mass 400 g.

Random samples of 6 tins are taken and the mean mass of each sample is recorded for quality assurance.

The production line is set up such that the sample means should be normally distributed with a mean of 400 g and a standard deviation of 3 g.

Using this information, draw warning lines on the control chart for the sample means.

Action lines have already been drawn on the chart.

A control chart for the sample ranges is also used and is shown below.

Explain why it is not appropriate to have lower warning lines and lower action lines on a control chart for sample ranges.

The first two samples of tins of beans have the following summary statistics.

Plot the summary statistics for these two samples on the control charts.

The tins in Sample 3 had the following masses:

405 g       408 g       407 g       406 g       403 g       408 g

Use these results to complete both control charts for Sample 3.

Describe what action needs to be taken after Sample 3.

Amy uses the Petersen capture recapture method in order to find an estimate for the number of macaques living in a plantation.

To do this, she catches an initial sample of 40 macaques from the plantation and she tags each one of them. She then releases the macaques back into the plantation.

For her second sample, Amy catches 55 macaques from the plantation. She finds that 8 of these macaques have tags.

Explain what needs to be true about her two samples for them to be valid for the capture recapture method.

Using the results, Amy is able to work out an estimate for the number of macaques living in the plantation.

Find Amy's estimate for the number of macaques and discuss the validity and the reliability of her estimate.

It is estimated that there are over 10 000 leatherback turtles who nest on the coast of Trinidad.

(Source: Institute of Marine Affairs)

Valerie travels to Trinidad to verify this estimate.

She captures a sample of 20 turtles, attaches a tag to each turtle and then releases them in the same area she captured them.

One year later, Valerie returns to the same area in Trinidad and catches a sample of 31 turtles.

She finds that 2 of these are tagged.

Valerie concludes that this information can be used to verify the estimate of 10 000.

Discuss the appropriateness of Valerie's method and of her conclusion.

As part of your discussion you should show your calculations and state any assumptions made.