Question

I work in construction and am pretty good in math, especially with fractions, so I didn't think I would ever have to write to you. But this one has got me stumped.

Last week I bought two boxes of nails, each from a different manufacturer. I poured the nails from one box into my left pocket and the other into my right. I started to use the ones in my left pocket, but they kept breaking and bending. I will never use them again. But here's my problem. I don't remember which box they came from.
I contacted both manufacturers and "Manufacturer A" said their boxes have a mean of 205.4 nails and a standard deviation of 1.8 nails. "Manufacturer B" said their boxes have a mean of 190.6 nails and a standard deviation of 3.3 nails. Both said their number of nails is normally distributed.
I counted the nails from my left pocket, including the used ones, and got 198. This is 7.4 nails less than Manufacturer A and 7.4 nails more than Manufacturer B. So I guess the standard deviation comes into play, but I don't know how.
Please explain which manufacturer most likely made these defective nails.
Sincerely,
Stumped Builder

Answer 1

This is an interesting question. Unfortunately we cannot say for sure which manufacturer has the faulty nails. But we can make an educated guess based on probability and how the normal distribution works.

Recall the z score conversion formula is:

z = (x - mu)/sigma

where

• x = raw score
• mu = mean
• sigma = standard deviation

Let's find the z score when x = 198 for manufacturer A.

z = (x - mu)/sigma

z = (198-205.4)/(1.8)

z = -4.11 approximately

This z score tells us how far we are from the mean in units of standard deviations. Something like z = -4.11 means we are 4.11 standard deviations below the mean. Often z scores beyond 3 standard deviations are fairly unusual/rare (though not impossible).

The value z = 0 is at the mean itself. The further we are from 0, the less likely chance of we're picking this manufacturer. The closer we are to z = 0, the more likely that we found the right box.

The majority of the nail counts clump around the mean. Then some boxes have larger counts, while others have smaller counts, and that produces the bell shaped curve. It might help to make a dot plot.

Now let's find the z score for manufacturer B

z = (x - mu)/sigma

z = (198-190.6)/(3.3)

z = 2.24 approximately

We find this z score is closer to z = 0 compared to z = -4.11

This tells us we have a higher chance of getting 198 nails from Manufacturer B compared to Manufacturer A. However, I must caution that this isn't a guarantee. There's a small chance that Manufacturer A might be the culprit.

Answer 2

I think that it is Manufacturer B, but read the following please.

Explanation:

The manufacturers claimed it is normally distributed, which is a bell shaped curve.

The center or peak of the curve is where the mean is.

Calculating the number of standard deviations from the mean for each corporation would likely give you your answer.

Manufacturer A

198 + 1.8 + 1.8 + 1.8 + 1.8 = 205.2

Manufacturer B

198 - 3.3 - 3.3 = 191.4

Manufacturer B seems to be about 2.2 SDs out from the mean, while Manufacturer A seems to be about 4 SDs.

I would think it belongs to Manufacturer B since it is closer, however I recommend waiting for another answer or doing further research.

On a TI-Inspire calculator, you can open a calculator and hit Menu, 6,5,2/3 (I think, under statistics to find normal CDF) Doing further research can help you find confidence of the value as it returns an output of confidence on a 1.0 scale.

Research standard deviation confidence intervals and try to logic this out based on what you are learning or studying.