How to determine the width of the zone of acceptance in hypothesis testing?

[amberglo]

First and foremost, I am terrible in Stats so bear w/ me on this one. This is my question, if anyone has an idea on how to figure out this problem that would be great! Here it is: Suppose you flip a coin 100 times and you want to test the hypothesis that the coin is fair, making sure there is less than a 5 percent chance of erroneously rejecting the fair coin hypothesis. How wide should the zone of acceptance be? How wide should the zone be if you flip the coin 5 times?

 

[CRGreathouse]

What you want is a range balanced around the mean which has cumulative probability 95%. The chance of 2 or 3 heads out of 5 flips is
((5 choose 2) + (5 choose 3)) / 2^5 = 20/32
The chance of 1 to 4 heads is
((5 choose 1) + (5 choose 2) + (5 choose 3) + (5 choose 4)) / 2^5 = 30/32

With 100 flips, you are probably intended to use the normal curve as an approximation rather than the explicit binomial formula, although either would work.

 

[tacman]

The width of the zone of acceptance in hypothesis testing is determined by the level of significance, which is typically denoted by the Greek letter alpha (α). In this scenario, the level of significance is 0.05 or 5%, as stated in the question. This means that there is a 5% chance of erroneously rejecting the fair coin hypothesis.

To determine the width of the zone of acceptance, we need to calculate the critical values for the test statistic. In this case, the test statistic is the number of heads that appear in 100 coin flips. The critical values can be found using a statistical table or by using a statistical software.

For a sample size of 100, the critical values are 40 and 60. This means that if the number of heads falls between 40 and 60, we accept the fair coin hypothesis. Any number outside of this range would lead to rejecting the fair coin hypothesis.

For a sample size of 5, the critical values are 1 and 4. This means that if the number of heads falls between 1 and 4, we accept the fair coin hypothesis. Any number outside of this range would lead to rejecting the fair coin hypothesis.

In summary, the width of the zone of acceptance depends on the level of significance and the sample size. In this scenario, the width of the zone of acceptance for a sample size of 100 is 20 (60 - 40) and for a sample size of 5 is 3 (4 - 1).