The range of values that can contain the population mean is based on the error threshold (Alpha Value).
Lets assume population mean is 16. As we have assumed that all good tyres are produced with 16 inches radius.
If we take a sample of 50 tyres, then we will have values like 16.2, 16.3, 15.98, 15.96, 15.99, 16.23…. so on an so forth.
For the sake of understanding let’s say the mean radius of those 50 tires came out to be 16.15. This is called Sample mean.
Now, based on this sample, we can calculate a range. The min and max values between which the mean of the population can be seen. The mean of the population is the mean of the radius of all the tires.
So basically, we are trying to estimate, how the mean of all the tyres look based on the given sample. And instead of giving a single value answer, we are providing a range of values. This range is known as Confidence Interval.
The confidence interval is affected by the alpha value. For every alpha value, we find the value of the statistic which gets multiplied with the standard error.
Confidence Interval = [ Mean(Sample) + N*(SE), Mean(Sample) + N*(SE)]
- SE=Standard Error=Standard Deviation of sample/sqrt(number of samples)
- N= Value of the statistic. If the population follows Normal Distribution then Z-statistic, if the population follow t-distribution then the t-statistic value for the given alpha value(probability of error margin)
For example, let us choose the alpha value of 5%. Hence, we are 95% confident that the mean value of the population will fall in between the confidence interval we find. Assuming normal distribution the value of N is 1.96 for alpha=5%. Similarly, the value of N is 2.68 for alpha=1%. So on and so forth. These “N” values are generated out of the probability distribution Z-values or the ideal bell curve distribution.
Hence, to calculate a confidence interval of the population mean. We need a sample of values, we calculate its mean, we calculate its standard deviation, we find the N-value based on the alpha level.
For the sake of explanation, assume below values were found for a sample of 50 tyres.
- Sample Mean of radius=16.15
- Standard deviation of 50 radius values=0.64
- n=50
- N=1.96 for alpha=5%
For the above values, the confidence interval will be calculated as [ 16.15 – 1.96*(0.64/sqrt(50)) , 16.15 + 1.96*(0.64/sqrt(50)) ].
Which comes out as [15.97 , 16.32].
Hence, based on the given sample of 50 Tyres we are 95% confident that the mean value of the radius of all the tires (population) will be somewhere between 15.97 and 16.32.