Normal Approximation-what is it?

[Quwee]

Suppose 20 people are randomly selected to participate in a survey. Assume that P(man)=P(woman)=.5. Use a normal approximation to estimate the probability that at least 12 women will be chosen.

 

[Unco]

Odd that you're asked about something you haven't learned yet.

Look in your textbook or google "normal approximation to binomial".

You'll need to apply a continuity correction (again, textbook/google), too.

 

[Quwee]

The teacher already taught all this stuff and I am behind the rest of the class
I read the textbook but is confusing and I don't understand it.

 

[Quwee]

I need to to do this so I can finish this page and then I can do the chapter test

 

[Unco]

Let X be the random variable for the number of women to be selected.

Binomial parameters:

p = 0.5
q = 1 - 0.5 = 0.5
n = 20

We are asked for P(X > 12), and we can use P(X > 12) = 1 - P(X < 12).

We're approximating a discrete distribution (binomial) with a continuous one (normal), so applying a continuity correction means we're looking for 1 - P(X < 11.5).

Normal distribution parameters:

\(\displaystyle \mbox{ \mu = np = (20)(0.5) = 10 }\) (np > 5 and nq > 5 requirements for the approximation are satisfied)
\(\displaystyle \mbox { \sigma = \sqrt{npq} = \sqrt{(20)(0.5)(0.5)} = \sqrt{5}}\)

\(\displaystyle \mbox{ Z = \frac{ X - \mu}{\sigma} = \frac{11.5 - 10}{\sqrt{5}} \approx 0.671}\)

A Z-score of 0.671 corresponds to a probability of 0.2489 in the normal tables.

Sketch the information on a bell curve to get the correct probability:

                          0.2489
                     ***   /
         0.5          :  |/_
            \   *     :      *
            _\|       :      :
           *          :      :   *
                      :      :  
    * - - - - - - - - - - - - - - - - - *   
                     10     11.5

So P(X < 11.5) = 0.5 + 0.2489 = 0.7489

And we have P(X > 12) = 1 - P(X < 12) = 1 - 0.7489 = 0.2511

If you have binomial tables which go up to n=20, although ultimately making the exercise pointless, you can compare the approximation with the actual.

 

[Quwee]

Thanks for your help, I was able to figure it out.