Section 5.5: Normal Approximation to the Binomial Distribution

The premise in our Larson textbook for using the normal approximation is that finding the binomial probability of an “less than x” or “greater than x” problem would be onerous – having to manually calculate each of the discrete values and then sum them up. They give an example of a doctor who performs a surgery that has an 85% success rate, but you want to know the probability that fewer than 100 of 150 attempted surgeries will be successful. And yes, finding each of the 101 discrete values (0 to 100) would be time-consuming if done manually using the equations in the book.

But, the reality today is that our software can easily do that – Excel, StatCrunch, and I think even the TI. Those software “technology” tools can find the cumulative probability from the left tail to the value of x we need. So, in the “real” world today, you would rarely, if ever, need to use the normal approximation to the binomial and its concomitant continuity correction.

Remember, the normal is an __approximation__ and not as precise as using the true binomial distribution.

That said, for this academic course, you do need to know how to use the normal approximation to the binomial at least for the homework. I have a blog post on the details of doing this for a problem from Section 5.5 here https://www.drdawnwright.com/?p=16678

I also have an online Excel calculator that makes this easy to do here https://www.drdawnwright.com/?p=17877

]]>A 95% confidence interval means that if you repeat exactly the same process/experiment many times, 95% of the time the population parameter, e.g. the mean, will fall within that interval. On a practical level, we can say that we are 95% confident the population parameter is within that interval.

One metaphor I like to use to help understand why a confidence interval gets wider as we increase the level of confidence is that of a basketball hoop. The standard hoop is 18 inches in diameter. If a player hits 90% of her free throws using a standard 18” rim, wouldn’t you think her success rate would increase as the size of the hoop increased beyond 18” and her success rate would decrease as the hoop got smaller than 18”? With a 20-inch diameter hoop, her success rate might increase to 95%. With a 24-inch diameter, her free throw success rate might be 99%. If the hoop were only 15 inches, her success rate might fall to 80%.

So it is with confidence intervals. The more confident we want to be, the wider the interval must be.

This is true because of the formula for the margin of error, E = z_{c}*σ/sqrt(n). Assuming the standard deviation σ and sample size n stay constant, E increases as the critical value of z increases. Z-critical increases as we increase the desired confidence level, c.

Here is how to find the critical value of z using StatCrunch. Granted, you will soon memorize the critical values for standard confidence levels of 90%, 95% and 99%, but you may be asked to find different values on quizzes and exams, e.g. 98% or 88%.

Use the steps for using the normal calculator to find z-scores but select the “Between” side (step 4). Enter the confidence level, c, in decimal format, e.g. here c= 95% is 0.95, in the window at step 5, and click compute. The red area is the 95% and the remaining 5% is split between the two tails. You will have the critical values of z, here +1.96 and – 1.96.

On the left, is the normal calculator set up to find the critical values for a confidence level of 88%. You should be able to see by comparing the two graphs that the 95% interval is wider than the 88% interval because the z-critical is larger for the 95%.

Hope this helps.

]]>She said in her discussion post:

My plan is to visit a haunted inn in Cobleskill, NY and stop by at the in-law’s house in Saratoga Spring, NY. I would also like to visit Salem, MA for historical sightseeing, and to stop by at the seafood restaurant in Mystic, CT.”

She chose to use a variation of the classic Traveling Salesman optimization problem and implemented it in Excel.

The result is shown below. Initially, the total distance was 606 miles. After using Solver, it was reduced to 586 miles. The order of cities was also rearranged. Solver suggests starting from Albany, Salem, Mystic, Cobleskill, and Saratoga. I was surprised that Cobleskill and Saratoga were the last cities to visit since they are so close to Albany. I believe the optimization models such as the Traveling Salesman Problem are beneficial for finding the minimum or maximum solutions. Without analyzing, I would have gone to visit Cobleskill and Saratoga first.”

Here is the sketch she created of her travel route.

Don’t you just love technology? And enterprising students?

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Over the last five-plus years that I have taught BUS233, I have noticed some characteristics of students who do well in the course. This is anecdotal evidence, for sure, but I think worth considering if you are facing 233 with a bit of trepidation.

Students who make a B or Better…

- Ask a lot of questions.
- Choose a stats technology quickly – usually StatCrunch – and invest the time to learn it.
- Spread their work in the course out over the week – they do not wait until Sunday afternoon to start.
- Make a detailed notebook of how they work problems.
- Learn the language of statistics – how to read word problems looking for keywords and phrases.
- Memorize a few key concepts (not equations) early in the term.

If I had to pick the most important, #1 is #1.

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