A particular type of plastic soap bottle is designed to have a capacity of 15 ounces. There is a variation in the bottle manufacturing process. Based on historical data, suppose that the bottle capacity can be reasonably modeled by a normal distribution with a mean of 15 ounces and a standard deviation of 0.2 ounces. What proportion of these bottles will have a capacity between 14.7 and 15.1 ounces?

well, I'm slightly confused as to what you are asking. I think you mean to ask what percentage of bottles will have a capacity between 14.7 and 15.1 ounces.

In order to do this, it is much easier to look at the problem in two parts:
1)the percentage from 14.7 to 15
2)the percentage from 15 to 15.1

1) to find the z score, subtract 15 from 14.7, and divide by 0.2. this should give you a z-score of -1.5. when you look at your chart that you should have in your book, you should find that the area for that z-score is 0.4332.
2) to find the z score, subtract 15 from 15.1 and divide by 0.2. this should give you a z score of 0.5. when you look at your chart, you should find that the are for that z score is .1915

you add the two areas together and you should get .6247. To make that a percentage, move the decimal point two places to the right and you should get 62.47%. This means that 62.47% of the bottles should fall between those two values.

hope that was helpful to you! :)