# Using approximations to the binomial distribution

• bonfire09
In summary, the conversation discusses using normal approximations to find the critical value for a two-tailed test involving a binomial distribution. It is clarified that the value for alpha should be checked for both one-tailed and two-tailed tests. The correct critical value for a two-tailed test with a proportion of 50% is determined to be 1.645.

## Homework Statement

This is the problem I am given.
. It is in he picture below or in the thumbnail. I was also told that since ##n## is big enough that I can use normal approximations.

## The Attempt at a Solution

I think that ##C_{\alpha}=C_{0.1}=2.33## which I got off the Z-score chart. The test statistic given looks like the one given for a binomial distribution given by where ##Z=\dfrac{x-np}{\sqrt{npq}}=\dfrac{\frac{x}{n}-p}{\frac{\sqrt{pq}}{\sqrt{n}}}##. I am not sure if this right or not. But it seems like this is the only way of finding the critical value. Thanks[/B]

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I think you looked up the value for alpha = .01, you were asked for .1. If you are using a normal approximation, then this is all you need to find the critical value. Be sure to clarify if this is a one-tailed or two-tailed test. From the question T>C_alpha indicates you are checking to see if a majority are in favor--i.e. one-tailed, your process is correct. If you were just checking whether or not the null hypothesis held, you would use a two-tailed test. In that case, you would need to divide alpha by two.

I see it is two tailed so I would use ##C=1.645## which would be my critical value.

Right--In this case the two tailed test is for p = 50%, if you reject that, then you know it is either more or less. If you were only concerned with the proportion being more, then you could use a smaller critical value to get the same level of confidence.

## 1. What is the binomial distribution?

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials with two possible outcomes (usually denoted as "success" and "failure") and a constant probability of success in each trial.

## 2. Why do we use approximations to the binomial distribution?

We use approximations to the binomial distribution because it can be difficult to calculate exact probabilities for large numbers of trials or when the probability of success is small. Approximations allow us to estimate these probabilities more easily and quickly.

## 3. What is the formula for approximating the binomial distribution?

The formula for approximating the binomial distribution is the normal distribution formula: P(x) = (1/σ√2π) * e^(-1/2 * ((x-μ)/σ)^2), where μ is the mean, σ is the standard deviation, and x is the number of successes.

## 4. When is it appropriate to use approximations to the binomial distribution?

Approximations to the binomial distribution are appropriate when the number of trials is large (usually greater than 30) and the probability of success is not too small (usually greater than 0.05). These conditions ensure that the normal distribution is a good approximation of the binomial distribution.

## 5. What are the limitations of using approximations to the binomial distribution?

The main limitation of using approximations to the binomial distribution is that they are only accurate for large numbers of trials and probabilities of success that are not too small. If these conditions are not met, the approximations may not provide accurate results. Additionally, the normal distribution may not accurately capture the shape of the binomial distribution in certain cases, leading to further inaccuracies.