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  Concept of Hypothesis test

Hypothesis Test: Pi = 3.2?

In the year 1897, legislature was introduced in Indiana that would make 3.2 the official value of pi for the State. Now, that sounds silly, however is it really?

Claim: Pi is 3.2.

To verify the claim, we are going to produce a whole bunch of values for pi, and then test to observe if the mean is 3.2.

H0: mu = 3.2 (that is, original claim)
H1: mu <> 3.2 (that is, two tail test)

Procedure:

Area of the unit circle is pi. In first quadrant the area of unit circle is pi/4. The calculator produces random numbers between 0 and 1. What we are going to do is to generate two random numbers that will simulate a randomly selected point in a unit square in first quadrant. If the point is in the circle, then the distance from (0,0) will be less than or equivalent to 1, and if the point is outside the circle, then the distance will be more than 1.

Encompass the calculator producing a squared distance from zero (that is, the square of distance explains the similar properties as far as being less than 1 or more than 1). Repeat this 25 times. Each time, record whether the point is within the circle (<1) or outside the circle (>1).

RAND^2 + RAND^2

Pi/4 is around equivalent to the ratio of points within the circle to the total number of points. Thus, pi will be 4 times the ratio of points within the circle to total number of points.

This entire process is repeated many times, and the mean and standard deviation is recorded.
The hypothesis test is then performed by using the t-test to see if the true mean is 3.2 (based on sample mean).

Illustration:

20 values for pi were produced by generating 25 pairs of arbitrary numbers and checking to see if they were within or outside the circle as explained above.

3.68     3.20     3.04     2.56     3.36
3.36     3.36     3.52     3.04     3.20
3.52     3.36     3.04     2.72     3.36
3.52     2.88     2.88     3.68     2.60

Mean of the sample is 3.194, that is, the standard deviation is 0.3384857923.

The test statistic t = (3.194 - 3.2)/(0.3384857293/sqrt(20)) = - 0.0792730931

The critical value, with 0.05 level of significance as none was stated, for a two-tail test by 19 degrees of freedom is t = +/- 2.093.

As the test statistic is not in critical region, the decision is fail to reject null hypothesis.

There is deficient evidence at 0.05 level of significance to refuse the claim that pi is 3.2.

It will be noted in the double speak, however it serves to exemplify the point. We would not dare to claim that pi was 3.2, even although this sample seems to exemplify this. The sample does not provide adequate evidence to show it is not 3.2; however there might be another sample somewhere that does give enough evidence. Therefore, we won't state it is 3.2; just that we do not have sufficient evidence to prove it is not 3.2.

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