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I'm reading about binomial distributions and I found this problem:

We're trying to find the probability of finding 70-245 successes, given a population of 245 and a probability of success of 25%. What is the probability of 70-245 successes summed?

In R the equation looks like this:

sum(dbinom(70:245, size = 245, p = 0.25)) and the answer is 0.113. Fine.

However, if I took a normal approximation, the answer is 0.0985. The mean and SD of the distribution is $$ u = np = 61.25 $$ and the $$ SD = 6.78 $$

When calculating the z score and finding the percentile, it falls a little bit short and it's 0.0985. Why is this?

The explanation I am reading is confusing:

Under the normal distribution, the probability of exactly 70 successes is undefined, so the shaded area above 70 doesn't included 70 successes. So we make a 0.5 adjustment and calculate the z score subtracting 0.5 from the observation of interest.

Why is 70 undefined? Why do we subtract 0.5?

  • 1
    Look up [continuity correction](https://en.wikipedia.org/wiki/Continuity_correction), perhaps? – 2017-02-16

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In order to catch the result of integer $70$ on the continuous Normal distribution, you need to evaluate over the range $69.5$ to $70.5$. So to model the chance of $70$ and above, your range needs to start at $69.5$.

The top end doesn't need adjustment because you only really want to use the Normal approximation when the end values are sufficiently unlikely; otherwise the binomial distribution will be not be well-fitted by the Normal.