# Hypothesis Testing For Binomial Distribution

I have come across the following problem which confused me quite a lot and would be helpful to get your comments:

Problem

Suppose we are testing a newly programmed random number generator. We use it to produce 1000000 random integers, each either 0 or 1. We get 1s for 500700 times and 0s for 499300 times. We'd like to test the null hypothesis $$H_0$$: this data comes from a binomial distribution $$b(1000000, \frac{1}{2})$$. against the alternative hypothesis $$H_1$$: nope.

First of all, what's the significance level $$p$$ of this test?

Then, discuss the accuracy of the following descriptions of our test:

• "We accept $$H_0$$ at the 0.05 significance level."
• "The probability that $$H_0$$ is true is $$p$$."
• "If $$H_0$$ id true, then the number of heads is 700 or more away from 500000 with probability $$p$$"
• "If $$H_0$$ is false, then the number of heads is 700 or more away from 500000 with probability $$(1-p)$$."
• "If $$H_0$$ is true, then the number of heads is less that 700 away from 500000 with probability $$(1-p).$$"

Things that confuse me:

• When the author asks: "What is the significance level $$p$$ of the test?" Do we have to calculate $$p$$ or should be just explain verbally that the significance level $$p$$ of this hypothesis test is the probability that we reject $$H_0$$ when $$H_0$$ is in fact true.

• Should I answer the first question in the following way?: "We never accept $$H_0$$ hypothesis at any significance level. We either reject it or not. We reject $$H_0$$ at 0.05 significance level if we observe the outcome of the experiment of which the probability of happening is less than 0.05 under $$H_0$$ and if cannot reject the $$H_0$$ if we do not observe such outcome, i.e. we do not have sufficient information against $$H_0$$ hypothesis."

• Should I answer the second question in the following way?: "We never say the $$H_0$$ is false with any probability. In the same way as for the previous question we either say that $$H_0$$ is false, or we cannot say that it is false."

• Should I answer the third question in the following way?: "This statement does not describe the test accurately, because if $$H_0$$ is true than the probability that the number of heads is 700 or more way from 500000 could be bigger or smaller than $$p$$. The probability that the number of heads is 700 or more away from 500000 under $$h_0$$ is actually $$p-value$$ of the test and it could be bigger or smaller than the significance level. If $$p-value$$ is bigger than the significance level then we cannot reject $$H_0$$ and if it is smaller we have sufficient evidence to reject $$h_0$$."