Like confidence intervals, if npq > 5, you can use the normal distribution as an approximation to the binomial for hypothesis testing. If we let po represent the proportion under the null hypothesis and qo = 1 - po, we can calculate the difference between the proportion in our sample p and po in standard deviations using:
You have decided to become a professional gambler specializing in roulette. If the roulette wheel is fair, each number has an equal chance (1/38) of coming up and the bets are structured so that you will lose in the long run. Your strategy is to find roulette wheels that are not properly balanced and produce some numbers more frequently than 1 out of 38 times. You believe that you have found just such a wheel and that it produces the number 29 more frequently than expected. You observe the wheel for 420 spins and note that the number 29 came up 14 times. How unlikely is this result?
The null and alterative hypotheses are:
Under the null hypothesis we have po = 1/38 = 0.0263 and qo = 0.9737. To see if we can use the normal approximation we calculate npoqo = 420(0.0263)(0.9737) = 10.76, which is greater than 5 so the normal approximation will work.
The observed proportion is 14/420 = 0.0333, which supports the alternative hypothesis, so we can continue. We can calculate how many standard deviations 0.0 333 is above 0.0263 as:
Using the starndard normal curve or 1 - normsdist(0.896) in Excel to get the probability (p-value) of 18.5%. If you are considering your life savings, then the probability that 29 comes up no more often than 1 out of 38 times is probably too high. If you are trying to decide whether to play this roulette wheel or go play blackjack, then it looks like this roulette wheel might be worth a shot. Again, the alpha value for a hypothesis testing problem should be set based on the problem situation and not on the data.