Large Sample Confidence Intervals

When you have large sample sizes, you can’t look up tables.  We use the approximation of the normal distribution instead to do our calculations.

Large sample confidence interval question

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Say I have 200 school students at a school.  50 of them need lunch provided for them at school.  If I have another identical group of 200 school students, find the 95% confidence interval in terms of how many will need lunch provided.


Now if I have a normal distribution curve, and I need a 95% confidence interval, it means that the middle area will be 95%.  The two outer areas, one on each side of the middle area, will be 2.5%. Note how this is a two-sided question – we are interested in the small 2.5% probability areas on both sides of the central peak. 

Firstly, we must find how many standard deviations from the middle the 2.5% areas are.  They are  from the centre, so we look up a standard deviation table.

They are 1.96 standard deviations from the centre.



First, let’s check we can use the normal distribution approximation:


For our sample, one standard deviation is:


And so, 1.96 standard deviations = 1.96 ´ 6.124 = 12.

·         Lowest possible number of lunches needed = 50 – 12 = 38.

·         Highest possible number of lunches needed = 50 + 12 = 62.

So, for the new 200 students, we could be 95% sure that at least 38 would need lunch, but no more than 62.