In 1897, legislature was introduced in Indiana which would make 3.2 the official value of pi for the State. Now, that sounds ridiculous, but is it really?
To test the claim, we're going to generate a whole bunch of values for pi, and then test to see if the mean is 3.2.
H0 : mu = 3.2 (original claim)
H1 : mu <> 3.2 (two tail test)
The area of the unit circle is pi. The area of the unit circle in the first quadrant is pi/4. The calculator generates random numbers between 0 and 1. What we're going to do is generate two random numbers which will simulate a randomly selected point in a unit square in the first quadrant. If the point is within the circle, then the distance from (0,0) will be less than or equal to 1, if the point is outside the circle, the distance will be greater than 1.
Have the calculator generate a squared distance from zero (the square of the distance illustrates the same properties as far as being less than 1 or greater than 1). Do this 25 times. Each time, record whether the point is inside the circle (<1) or outside the circle (>1).
RAND^2 + RAND^2
Pi/4 is approximately equal to the ratio of the points inside the circle to the total number of points. Therefore, pi will be 4 times the ratio of the points inside the circle to the total number of points.
This whole process is repeated several times, and the mean and standard deviation is recorded.
The hypothesis test is then conducted using the t-test to see if the true mean is 3.2 (based on the sample mean).
20 values for pi were generated by generating 25 pairs of random numbers and checking to see if they were inside or outside the circle as illustrated above.
The mean of the sample is 3.194, the standard deviation is 0.3384857923.
The test statistic t = (3.194 - 3.2) / (0.3384857293/sqrt(20)) = -0.0792730931
The critical value, with an 0.05 level of significance since none was stated, for a two-tail test with 19 degrees of freedom is t = +/- 2.093.
Since the test statistic is not in the critical region, the decision is fail to reject the null hypothesis
There is insufficient evidence at the 0.05 level of significance to reject the claim that pi is 3.2.
Note the double speak, but it serves to illustrate the point. We would not dare to claim that pi
was 3.2, even though this sample seems to illustrate this. The sample doesn't provide enough
evidence to show it's not 3.2, but there may be another sample somewhere which does provide
enough evidence (let's hope so). So, we won't say it is 3.2, just that we don't have enough
evidence to prove it isn't 3.2.