# In this investigation my aim is to find the mean height of boys and girls in year 10.

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Introduction

In this investigation my aim is to find the mean height of boys and girls in year 10. I also want to find the variance and standard deviation of the girls and boys height. I also want to find 90%, 95% and 99% confidence intervals of the girls and boys mean height. If the sample size is large enough, the distribution of the sample mean is approximately Normal. The variance of the distribution of the sample mean is equal to the variance of the sample mean divided by the sample size. These are true whatever the distribution of the parent population. The Central Limit Theorem allows predictions to be made about the distribution of the sample mean without any knowledge of the distribution of the parent population, as long as the sample is large enough. For this reason, the sample size will be set at 30, which I consider large enough for the distribution of its mean to be normal (according to the Central Limit Theorem). It should not be larger because the aim of this investigation is to carry out a “small scale survey”

The variance is a measure of how spread out a distribution is. It is computed as the average squared deviation of each number from its mean.

Middle

Ф = area to the left

95% confident

0 Z1 0.025

0.95

Ф Z1 = 0.975

So Z1 = 1.96 (from table)

99% confidence

0 Z1 0.005

099

Ф Z1 = 0.995

So Z1 = 2.575 (from table)

To find the confidence intervals you need to know

- standard deviation of the sample
- mean of the sample
- sample size
- standard error

How to calculate the confidence intervals.

µ–Z1 (S.E)> x< µ +Z1 (S.E)

Separate them:

µ< x + Z1 (S.E)

µ> x– Z1 (S.E)

Then put mean in middle stating what it’s more than and less than

……<µ<……

I chose a sample because it is impossible to weigh the whole population. The sample must be random for the Central Limit Theorem to be in effect, so that the distribution of its mean is Normal and predictions can be made about it, even though the distribution of the parent population of heights is unknown and not necessarily Normal.

The population I have chosen for this investigation is the year ten heights in Leicester. I wrote all the names of the schools in Leicester on separate pieces of paper and put them into a hat. I shook the hat and randomly picked one school out. The school that was picked was Beauchamp College.

Conclusion

The z1 score for 99% is 2.575. To obtain the confidence interval (such as the ones calculated above), I would have to multiply this figure and by the s.e. and add it to or subtract it from the sample mean. However, now I have the confidence interval, and I am trying to work out the correct size of the sample so that the standard error is small enough to have a very small confidence interval. In essence I am doing the process backwards.

The size of the sample was small. The calculations that relied upon the data collected are therefore inaccurate to some extent. To be more accurate a large sample must be collected. Accuracy in the realm of to 0.001g is unlikely to be needed, so a larger sample would not necessarily have to be as big as 20,000, which is very impractical

The sample might have been a “fluke” I might have got all the tall girls and boys or all the short ones. However there is not much to do to eliminate the possibility of this apart from to measure every single every boy and girl. This is extremely impractical (possibly impossible).

The children measured were from one area, they do not represent all the children in the world, only ones in that area.

This student written piece of work is one of many that can be found in our AS and A Level Probability & Statistics section.

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