# Math Help - Normal Distribution Questions

1. ## Normal Distribution Questions

1) Hertz Brothers, a small, family-owned radio manufacturer, produces electronic components domestically but subcontracts the cabinets to a foreign supplier. Although inexpensive, the foreign supplier has a quality control program that leaves much to be desired. On the average, only 80% of the standard 1600-unit shipment that Hertz receives is usable. Currently, Hertz has back orders for 1260 radios but storage space for no more than 1310 cabinets. What are the chances that the number of usable units in Hertz's latest shipment will be large enough to allow Hertz to fill all the orders already on hand, yet small enough to avoid causing any inventory problems?

2) There is a theory embraced by certain parapsychologists that hypnosis can enhance a person's ESP ability. To test that hypothesis, an experiment was set up with 15 hypnotized subjects. A total of 326 correct identifications were made. Can it be argued on the basis of those results that hypnosis does have an effect on a person's ESP ability? Explain.

3) Suppose that 100 fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Include a continuity correction in your analysis.

4) Considerable controversy has arisen over the possible aftereffects of a nuclear weapons test conducted in Nevada in 1957. Included as part of the test were some 3000 military and civilian "observers." Now, more than 40 years later, eight cases of leukemia have been diagnosed among those 3000. The expected number of cases, based on the demographic characteristics of the observers, was three. Assess the statistical significance of those findings. Calculate both an exact answer using the Poisson distribution as well as an approximation based on the central limit theorem.

2. Not a single word how you plan to answer these questions?

You WILL have to do better than that.

3. Sorry, the questions were long, so I just left it at that because I didn't want the post to be really long.

1) I have to find the probability that the number of units is between 1260 and 1310. Normally, 1280 units make it (this is the mean).
How do I know the standard deviation though? I'm probably over thinking this part.

2) This needs further clarification that's in my book, but I think I figured this one out anyway.

3) Mean would be 350 right? I think my problem is the same as 1) in that I'm over thinking the standard deviation part and it's rather simple.

4) This one I'm not really sure. Mean is 3 cases, and I have to figure out the probability that 8 cases came up. Obviously if the probability is really low (which I'm expecting it is) then this would mean the nuclear weapons test made the observers more susceptible to leukemia.

4. Not bad. Let's talk about #1. A couple of ways come to mind for the variance.

1) Emperical Rule. 1600 is the maximum, so let's call that 3 Standard Deviations. (1600-1280)/3 = 320/3 = 107. -- That seems a little large.

2) Binomial Approximation: p = 0.8, q = 0.2, n = 1600, Variance = 1600*0.8*0.2 = 256. Standard Deviation = 16. -- That seems a little small. Maybe not.

Other ideas?

5. Oh wait.... is it sqrt(np(1-p))? I must have just missed this the first time I went through the chapter in the book but looking at it now it seems that's what it is.

6. Since you told us we needed to use a Normal Distribution, that's kind of a stretch, but that is what I call the Binomial Approximation up above.

7. Hmmm... well I think I'm OK with #1 now. #4 still confuses me though, especially the Poisson part. I know the basics of the Poisson distribution but I'm confused as to how I figure out the exact answer for the probability.

8. That's the easiest one! Try $\lambda = 3$ and calculate p(8) = ??