Question

A die is tossed 600 times. H0 is the hypothesis that the proportion of tosses showing aces is binomially distributed with mean 1/6. Find the upper limit of the region for which H0 is accepted at the 1% level of significance in a two sided test.

Answer 1

In this case null hypothesis represents the proportion of tosses showing aces which is binomially with mean 1/6.

let p represents the probability of success in tossing a die, which is the proportion of aces. Then

Let n be the number of times die rolled. Here n=600.

Let X be the random variable which represents the number of times aces comes when die is rolled. If it is binomially distributed then the probability mass function is given by

If X is Binomially distributed, then the mean and standard deviation is given by

Therefore if the die is 600 times rolled and p is 0.166. Then the mean and standard deviation becomes

For obtaining the upper limit, we first obtain the confidence interval for level of significance =0.01.

Let us consider Null and Alternative hypothesis as follows

Then the confidence interval is given by

From the above mean and standard deviation, we have

Then the confidence interval for the given data for the mean under accepted is given by

From the confidence interval of (99.05, 100.95), we can see that the upper limit of the interval for the null hypothesis accepted is 100.95.