How does the spread of a set of sample means compare with the spread of the population they came from?

The sample means are more closely distributed than the individual values in the population. Each mean lies between the smallest and largest values of its own sample, so the means can never reach as far out as the extremes of the population.

Why is quality assurance based on samples rather than on every item?

Samples are taken at regular intervals and their means, medians or ranges compared against target values. It is usually impossible for every manufactured item to be exactly correct, so what matters is that they do not vary too much from the target.

Define control chart .

A control chart is a time series chart used in quality assurance, with the sample number along the horizontal axis and the sample mean, median or range up the vertical axis. Horizontal lines mark the target value together with the warning and action limits.

A point on a control chart falls between the upper warning line and the upper action line. What should happen?

Take another sample straight away to check whether there is a problem. A point in that band might be nothing more than random variation, but it might not, so it is neither ignored nor acted on until it has been confirmed.

A point on a control chart falls above the upper action line. What should happen?

Stop the manufacturing process and reset the machinery to bring it back within the limits. A point beyond an action line almost certainly means something has gone wrong, rather than being random variation.

Why is a control chart for sample range usually drawn with only upper limits?

Because the target range is zero, which would mean every item exactly on target with no variation at all. A range can only be too large, never too small, so a lower limit would serve no purpose.

What problem can a sample range chart catch that a sample mean chart misses?

Items varying wildly either side of the target, since a very large and a very small value cancel each other out in a mean. The mean would look perfectly acceptable while individual items were far from what the customer should be getting.

True or False? For a properly working process, about 1 in 20 sample means will fall outside the warning limits.

True. Sample means are normally distributed, so about 95% of them fall within the warning limits. That leaves roughly one in twenty outside even when nothing at all is wrong, which is why a warning is checked rather than acted on.

What lets you use a sample statistic as an estimate for the population?

The sample being representative of the population. If it is, the population's mean, median, quartiles and range can all be taken as approximately equal to the sample's.

True or False? Two different samples from the same population will give the same statistics.

False. Statistics from a sample will usually not match the population exactly, and two different samples from the same population will not usually match each other either. That is simply how sampling behaves, and it is why a sample statistic is called an estimate.

How would you make an estimate from a sample more reliable?

Use a larger sample. A bigger sample is more likely to be representative of the population, so the statistics calculated from it come out closer to the population's own.

Define the capture recapture method.

The capture recapture method estimates the size of a population that cannot practically be counted, such as the fish in a lake. A first sample is caught, marked and released, then a second sample is caught later to see what proportion of it carries a mark.

Name two assumptions the capture recapture method depends on.

That the tagged members had time and opportunity to mix back in with the rest of the population before the second sample was taken. And that the population stayed about the same size in between, with no significant births, deaths or migration, and no tags lost.

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Quality Assurance & Estimation (AQA GCSE Statistics: Higher): Flashcards

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How does the spread of a set of sample means compare with the spread of the population they came from?

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How does the spread of a set of sample means compare with the spread of the population they came from?
The sample means are more closely distributed than the individual values in the population.
Each mean lies between the smallest and largest values of its own sample, so the means can never reach as far out as the extremes of the population.
Why is quality assurance based on samples rather than on every item?
Samples are taken at regular intervals and their means, medians or ranges compared against target values.
It is usually impossible for every manufactured item to be exactly correct, so what matters is that they do not vary too much from the target.
Define control chart.
A control chart is a time series chart used in quality assurance, with the sample number along the horizontal axis and the sample mean, median or range up the vertical axis.
Horizontal lines mark the target value together with the warning and action limits.
Where are the warning and action limits drawn on a control chart for sample means?
The warning limits go two standard deviations either side of the target mean, at $μ \pm 2 σ$ on the chart.
The action limits go three standard deviations either side, at $μ \pm 3 σ$ for that process.
A point on a control chart falls between the upper warning line and the upper action line. What should happen?
Take another sample straight away to check whether there is a problem.
A point in that band might be nothing more than random variation, but it might not, so it is neither ignored nor acted on until it has been confirmed.
A point on a control chart falls above the upper action line. What should happen?
Stop the manufacturing process and reset the machinery to bring it back within the limits.
A point beyond an action line almost certainly means something has gone wrong, rather than being random variation.
Chocolate bars have a target mean of 61.4 g with a standard deviation of 0.8 g. Where do the warning lines go?
At $61.4 + 2 \times 0.8 = 63.0$ grams above the target and $61.4 - 2 \times 0.8 = 59.8$ grams below it.
Both lines are drawn horizontally right across the chart and clearly labelled.
Why is a control chart for sample range usually drawn with only upper limits?
Because the target range is zero, which would mean every item exactly on target with no variation at all.
A range can only be too large, never too small, so a lower limit would serve no purpose.
What problem can a sample range chart catch that a sample mean chart misses?
Items varying wildly either side of the target, since a very large and a very small value cancel each other out in a mean.
The mean would look perfectly acceptable while individual items were far from what the customer should be getting.
True or False?
For a properly working process, about 1 in 20 sample means will fall outside the warning limits.
True.
Sample means are normally distributed, so about 95% of them fall within the warning limits.
That leaves roughly one in twenty outside even when nothing at all is wrong, which is why a warning is checked rather than acted on.
What lets you use a sample statistic as an estimate for the population?
The sample being representative of the population.
If it is, the population's mean, median, quartiles and range can all be taken as approximately equal to the sample's.
A sample of 50 rabbits has a total weight of 82.5 kg. Estimate the mean weight of the whole population.
The sample mean is $\frac{82.5}{50} = 1.65$ kilograms per rabbit.
Assuming the sample is representative, the population mean is taken to be about the same.
A sample gives a lower quartile of 1.2 kg and an upper quartile of 2.1 kg. Roughly how many of a population of 600 weigh between those?
About 300, since half the data lies between the quartiles and $\frac{1}{2} \times 600 = 300$ of them.
The same reasoning gives about a quarter of the population below the sample's lower quartile.
True or False?
Two different samples from the same population will give the same statistics.
False.
Statistics from a sample will usually not match the population exactly, and two different samples from the same population will not usually match each other either.
That is simply how sampling behaves, and it is why a sample statistic is called an estimate.
How would you make an estimate from a sample more reliable?
Use a larger sample.
A bigger sample is more likely to be representative of the population, so the statistics calculated from it come out closer to the population's own.
Define the capture recapture method.
The capture recapture method estimates the size of a population that cannot practically be counted, such as the fish in a lake.
A first sample is caught, marked and released, then a second sample is caught later to see what proportion of it carries a mark.
Complete the Petersen capture recapture formula.
$N = \frac{M \times______}{______}$
The completed formula is:
$N = \frac{M n}{m}$
Here $N$ is the population, $M$ the first sample size, $n$ the second sample size and $m$ the number marked in the second sample.
50 rabbits are tagged and released. Later 100 are caught and 8 of them are tagged. Estimate the population.
Substitute into the capture recapture formula with $M = 50$ and $n = 100$ and $m = 8$ for this study.
$N = \frac{50 \times 100}{8} = 625$
Name two assumptions the capture recapture method depends on.
That the tagged members had time and opportunity to mix back in with the rest of the population before the second sample was taken.
And that the population stayed about the same size in between, with no significant births, deaths or migration, and no tags lost.

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