Statistics Help?

Question

I asked this question in the general form earlier, but the response wasn't what I needed; so here I am, asking it in its specific form.

Note: this ISN'T my homework problem - this is the example problem of the same style that I know the answer to, but I can't figure out how they got the answer.

Consider a random sample X1 ... Xn of normally distributed random variables with mean u and variance = 7.

If n=15, what is the probability that |u - x|<=.4? What if n = 50?

I know the answers to this are .4418 and .7150, but I can't figure out how they got that. I understand how to work backwards from having the probability and finding the c (.4 in this problem), but I can't figure out how to bring it the other way.

Answer 1

The standard deviation of the mean of a population of size n taken from another population with variance 7 is 7/n. So if you take a sample of 15, the variance of the mean distribution is 7/15. The standard deviation is sqrt(7/15). From this, you find the z score of 0.4 (I get .586) and look up the probability. Using Excel to do the z score for me, I get .4418. Use the same technique for 50.

Put another way, let's say you pull 15 samples and take the mean 100 times. Those means will also form a normal distribution, but you would expect the means to be closer together (smaller variance) than the distribution you were pulling from (central limit theorem).

Answer 2

Here's the approach, you can work the numbers.

You mention that the data is normally distributed. This means that the shape of the distribution curve itself is a bell curve.

The area under the bell curve is 1.0. So, you need to integrate the area under the curve between the limits |u-x|<=0.4. This will be some percentage of the total area, and is the probability that you are looking for.

Now, how to do it...

You need to start with the general equation for a bell curve. It

Check out the equation 'Probability density function" found at the following link (its easier to point you there than to type it)

http://en.wikipedia.org/wiki/Normal_distribution

You will see that the shape of the curve is dependent on 'u', the mean, and sigma, the standard deviation. It is not dependent on N, the number of samples. This is because the curve is based on the assumption of a large sample size (typically N>30).

The standard deviation is the square root of the variance. You mention that the variance is 7, so the std. deviation would be 2.645.

For small samples, you can use the T-distribution (sometimes called the Student T test). This function has is a wider Gaussian shape indicating a higher uncertainty due to small sample sizes.
The curve is different but the process is the same. Integrate under the curve between the limits you are interested in, and ratio it to the total area under the curve.

Hope this helps..