Using StatCrunch To Present Normal Approximation To The Binomial Distribution

March 17, 2010 at 12:39 pm Leave a comment

Today I introduced the normal approximation to the binomial distribution to my class. I used StatCrunch to help with the presentation and it went really well.

Beginning Example

Since it has been a week or so since we were working with the binomial distribution, I put the following problem on the board to discuss.

Example 1: 60% of students at our college are female. If we randomly select 10 students, find the probability that 7 are female.

Students identified that it was a binomial problem, and we talked about how to identify a problem as a binomial problem. We then talked about why using the binomial formula was an effective strategy for solving this problem. I then edited the problem as shown.

Example 2: 60% of students at our college are female. If we randomly select 10 students, find the probability that at least 7 are female.

Students commented that by rewriting 7 as “at least 7” made the problem a little more difficult, but we could then use the binomial tables and add the probabilities associated with x=7 through x=10. So I edited the problem again.

Example 3: 58% of students at our college are female. If we randomly select 10 students, find the probability that at least 7 are female.

Changing p from 0.6 to 0.58 meant that the binomial tables were no longer an option. The students understood that they would need to use the binomial formula for x=7 through x=10. Although it was a little inconvenient, the problem could still be handled in a relatively easy manner. That’s when I changed the problem one more time.

Example 4: 58% of students at our college are female. If we randomly select 100 students, find the probability that at least 70 are female.

OK, now we have a problem. We cannot use the binomial tables, and using the formula would require us to use 31 different values of x. Using the complement would require us to use 70 different values of x.

Enter StatCrunch

I opened up StatCrunch and went to the binomial calculator (Stat > Calculators > Binomial).

We could see that technology could give us a solution rather quickly. The probability is 0.0090. But we also saw something that was very valuable. My students saw that the probability histogram was approximately normal. They were thinking that it was too much of a coincidence. I commented that it was too bad we didn’t know the mean or standard deviation for this distribution, and a student quickly reminded me that we did have a formula for those.

Continuity Correction

One stumbling block for past students is how to apply a continuity correction, or even understanding why. With the probability histogram still on the board I could clearly point out the gaps between the possible discrete values of x, and marked how we split those gaps in half. I also was able to explain why we made our adjustment downward in this example because the histogram was already red for every value beginning at 70, but we needed to drop down to 69.5 (the lower limit for x = 70).

Normal Approximation

We walked through the steps and students saw that we could get a reasonably good approximation of the probability without having to use the formula 31 times. The students appreciated this efficient estimation, and truly seemed to understand what was going on rather than simply mimicking my work.

Conditions To Apply The Normal Approximation

The condition to apply the normal approximation is given in our textbook as np(1-p)>10. Using StatCrunch I was able to show how the probability histogram becomes more bell-shaped as the value of n increases.

n=10


n=20  

n=50

n=100

n=1000

Conclusion

All in all, a pretty productive day with a classroom full of students who truly understood.

-George

I am a mathematics instructor at College of the Sequoias in Visalia, CA. I blog about general teaching ideas every Wednesday. Let me know if there are other topics you’d like me to cover. You can email suggestions through the contact page on my website.

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