Row data.
WHAT CALCULATIONS WILL BE MADE USING THE DATA.
- The mean, standard deviation and variance of the sample.
- These will be used to estimate the variance and standard deviation of the parent population.
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This will also be used to create confidence intervals for the mean of the parent population.
- Also, calculations that determine the size that a possible sample could be to achieve a certain percentage confidence interval for the mean to be a certain range.
First I have to sort all the responses I got in numerical order so that I can use a random number generator to get my sample of 100 responses.
Using a random number generator, I picked 100 random responses as my sample.
RESULTS (SAMPLE DATA):
To calculate the mean of the variance for the total number of 13-16 year old students who got pockect money on weekly basis:
Let p be the proportion of student who get £5 or more.
Although p is not yet know a good estimate will be: 54
100
Let X be the number of people in the sample who said they get more than £5 for their pocket money.
Then X~(100,p)
∴ to find mean for the sample (X) =np
= 100xp
=100p
Variance = npq
=100xpx(1-p)
=100p(1-p)
Provided that n, the size of sample is large enough (n ≥ 30), I can approximate this distribution with a Normal Distribution having the same mean and variance:
X ~ N(μ, σ2)
X ~ N(100p, 100p(1-p))
(but p = 0.54)
Mean = 100x0.54 μ =54
Variance =100x0.54x0.46 =24.84 using the formula : n s2 σ2 = 25.09
n - 1
CONFIDENCE INTERVALS:
To calculate how confident I am about the estimate of the population mean, I can use confidence intervals. These tell me how confident (as a percentage) I can be that the mean of the population falls within a given range.
Using the formula :
ps – 1.645 ps(1 – ps) < p < ps – 1.645 ps (1 – ps)
n n
I will be able to find 90% confidence interval for the proportion of the population
(p) who get £5 or more for their pocket money.
0.54 – 1.645(√(0.54x0.46)/100) < p < 0.54 + 1.645(√(0.54x0.46)/100)
0.54 – 0.0812 < p < 0.54 + 0.0812
0.4588 < p < 0.6212
between 45.9% and 62.1% (to 2 s.f.)
In other words, I am 90% cofidence that between 45% and 62% of the 374 students who answered the questionnaire get £5.00 or more for their pocket money.
To find 95% the confidence interval:
0.54 – 1.96(√(0.54x0.46)/100) < p < 0.54 + 1.96(√(0.54x0.46)/100)
0.54 – 0.098 < p < 0.54 + 0.098
between 44.2% and 63.8% get £5.00 or more for their pocket money.
FURTHER CONFIDENCE INTERVAL:
If I wanted to be 99% confident;
0.54 – 2.33(√(0.54x0.46)/100) < p < 0.54 + 2.33(√(0.54x0.46)/100)
0.54 – 0.116 < p < 0.54 + 0.116
this would mean between 42.4% and 65.6% students who answered the questionnaire would get £5.00 or more for their pocket money.
PARAMETERS FOR THE SECOND POPULATION:
To find the proprtion of 13 to 16 year old students in the Suffolk County who normally get £5.00 or more for their pocket money, I will use all the responses I got as my sample; rather than taking a sample from it.
For this population random sampling is very hard to collect since this would involve questioning all 13 to 16 year olds in each individual school in the Suffolk County, and then randomly select my sample.
For this reason I had to uses all of the responses as my sample.
To find the mean = np
= 374 x (230/374)
μ= 230
s2 = 374 x (0.615 x 0.385)
= 88.6
using n/(n – 1)s2 ; we have 374/(374-1) x 88.6
σ2 = 88.8
Caluculating 90% confidence interval:
0.615 – 1.645(√(0.615x0.385)/374) < p < 0.615 + 1.645(√(0.615x0.385)/374)
0.574 < p < 0.656
∴ between 57.4% and 65.6%
To find the 95% confidence interval:
0.615 – 1.96(√(0.615x0.385)/374) < p < 0.615 + 1.96(√(0.615x0.385)/374)
0.566 < p < 0.664
i.e between 56.6% and 66.4%
CONCLUSION:
Comparison of Cofidence Intervals
Looking at the two tables I can clearly see that the confidence intervals of the first popultion are bigger than those of the second population. This is because as I increased my sample size from 100 (first population) to 374 (second population), the value of the mean was getting closer to its true value.
Going back to my original question “HOW MUCH POCKET MONEY DO YOU NORMARLLY GET ON WEEKLY BASIS?: OVER £5 OR UNDER £5” I can say that I am 95% confident that between 44% and 64% of all the student who participated in the qustionnaire get £5.00 or more every week for their pocket money.
I am also 95% confident that between 57% and 66% of all 13 to 16 year olds in the Suffolk County receive £5 or more for their weekly pocket money.
POSSIBLE EXTENSION:
If I had enough time, I would also carry out the short survey to other schools within Suffolk.
This would improve on my sampling of the second population and also would give me accurate population parameters; rather than using results from one school as the sample.
LIMITATIONS:
The size of the sample of the first population was small (compared to the popultion itself) ; therefore calculations that relied upon the data collected are therefore inaccurate to some extent.
The results may be unreliable because some of the student might have given false information i.e lie about the amount of money they are given by their parents for pocket money.
This could mean that the actual population parameters are somewhat different to the ones estimated here.