Question 1037032
mean = 12450
s = 570.


you want to be confident that a random cartridge sold will provide more than the number of pages quoted 90% of the time.


this means that only 10% of the cartridges sold will provide a number of pages less than that.


you would use a z-score calculator to determine the z-score that has the probability of being less than the indicated z-score only 10% of the time.


that z-score will be -1.29 with an accuracy to the nearest 2 decimal places.


i used a z-score calculator that says that the actual z-score is -1.2815515767.


that would round to -1.28, but i chose -1.29 for the reasons that will be explained further below.


if you look up the the z-score in the z-score table, it will tell you that .....


a z-score of -1.28 has .1003 proportion of the normal distribution curve to the left of it.


a z-score of -1.29 has .0985 proportion of the normal distribution curve to the left of it.


you could use the calculator, or you could interpolate, or you could take the conservative approach and use -1.29 as your z-factor.


i chose -1.29 because this will ensure you that, at least 90% of the cartridges selected will have a number of pages greater than that.


the actual number could be higher, but it is highly improbable that it will be lower.


given a z-score of -1.29, you would calculate the raw score as follows:


z = (x - m) / s


z is the z-score.
x is the raw score.
m is the mean.
s is the standard deviation.


the formula becomes -1.29 = (x - 12450) / 570.


solve for x to get x = -1.29 * 570 + 12450 = 11714.7.


you can predict that any cartridge that was sold will be able to print at least 11715 pages 90% of the time.


there is a graphing calculator that will show you this.


that calculator can be found at <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">http://davidmlane.com/hyperstat/z_table.html</a>


plug in the numbers and the calculator tells you that you can achieve a raw score greater than 11715 for approximately 90.14% of the time.


here's an output from the calculator.


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