Normal approximation (of sample average distribution)

1) determine what c is in terms of the standard deviation of the normal distribution

Assume zero mean for this, it has no effect
70% mass inside [-c,c] means a lower tail of 1/2(100-70) = 15% mass.

The inverse CDF value for 0.15 is a z-value of -1.0364334.
This means that c = 1.0364334 times the standard deviation of the underlying normal distribution.

2) Determine what the standard deviation of the sample average is.

When approximating a sample distribution by the normal distribution the variance used is the variance of the underlying distribution
divided by the number of samples. The standard deviation will be the square root of this.

The underlying distribution has a mean of 26 days
In an exponential distribution the variance is the square of the mean, i.e. 676 days

So the sample variance is [MATH]\dfrac{676}{80} = \dfrac{169}{20}[/MATH]and the sample standard deviation [MATH]\sigma = \dfrac{13}{2\sqrt{5}}[/MATH]
[MATH]c = 1.0364334 \cdot \sigma \approx 3.012796[/MATH]
 
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