## An Engineer Is Going To Redesign An Ejection Seat For

Summary: This text discusses the concept of z-scores and their use in probability calculations. It includes tables and calculators for converting z-scores and probabilities, and emphasizes the importance of z-scores in understanding how a score compares to a group of scores. The text also provides examples of using a normal distribution calculator to find probabilities.

To find the probability that a pilot's weight is between 140 lbs. and 190 lbs. given a normally distributed population with a mean of 158 lbs. and a standard deviation of 27.2 lbs, we can use the z-score formula and then find the probabilities using a standard normal distribution table or calculator.

The z-score, which measures how many standard deviations a value is from the mean, can be calculated using the formula:

[ Z = \frac{(X - \mu)}{\sigma} ]

Where:

- (X) = 140 lbs (lower limit of weight range)
- (\mu) = 158 lbs (mean weight)
- (\sigma) = 27.2 lbs (standard deviation)

This gives us: [ Z_{140} = \frac{(140 - 158)}{27.2} ] [ Z_{140} = \frac{-18}{27.2} ] [ Z_{140} = -0.662 ]

Similarly for the upper limit: [ Z_{190} = \frac{(190 - 158)}{27.2} ] [ Z_{190} = \frac{32}{27.2} ] [ Z_{190} = 1.176 ]

Now, we need to find the probability that a z-score is between -0.662 and 1.176. This can be obtained using a standard normal distribution table or calculator.

By looking up these z-scores in the standard normal distribution table or using a calculator, we find that the probability of a pilot's weight being between 140 lbs. and 190 lbs. is approximately 0.5594, or 55.94%.

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