The Empirical Rule is just an approximation and only functions for certain worths. What if you desire to uncover the probcapability for x values that are not integer multiples of the conventional deviation? The probcapability is the area under the curve. To uncover locations under the curve, you need calculus. Before innovation, you necessary to convert eexceptionally x value to a standardized number, dubbed the z-score or z-worth or simply just z. The z-score is a measure of exactly how many type of conventional deviations an x worth is from the expect. To convert from a normally dispersed x worth to a z-score, you usage the complying with formula.

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Definition (PageIndex1): z-score

wright here (mu)= intend of the population of the x value and (sigma)= traditional deviation for the populace of the x value

The z-score is normally dispersed, through a suppose of 0 and also a conventional deviation of 1. It is well-known as the typical normal curve. Once you have actually the z-score, you can look up the z-score in the typical normal distribution table.

Definition (PageIndex2): typical normal distribution

The conventional normal distribution, z, has a expect of (mu =0) and a traditional deviation of (sigma =1).

Figure (PageIndex1): Standard Regular Curve

Example (PageIndex1) general normal distribution

The size of a human pregnancy is usually spread through a expect of 272 days via a traditional deviation of 9 days (Bhat & Kushtagi, 2006).

State the random variable. Find the probability of a pregnancy lasting even more than 280 days. Find the probcapability of a pregnancy lasting less than 250 days. Find the probcapability that a pregnancy lasts between 265 and 280 days. Find the size of pregnancy that 10% of all pregnancies last less than. Suppose you satisfy a womale that says that she was pregnant for much less than 250 days. Would this be inexplicable and what could you think?


a. x = length of a human pregnancy

b. First translate the statement into a mathematical statement.

P (x>280)

Now, attract a photo. Remember the center of this normal curve is 272.

Figure for Example (PageIndex1)b

To uncover the probcapacity on the TI-83/84, looking at the image you realize the lower limit is 280. The upper limit is infinity. The calculator doesn’t have actually infinity on it, so you need to put in a really huge number. Some people favor to put in 1000, but if you are functioning through numbers that are bigger than 1000, then you would need to remember to readjust the top limit. The safest number to use is (1 imes 10^99), which you put in the calculator as 1E99 (wbelow E is the EE switch on the calculator). The command also looks like:

( extnormalcdf(280,1 E 99,272,9))

Figure (PageIndex3): TI-83/84 Output for Example (PageIndex1)b

To uncover the probcapability on R, R always gives the probability to the left of the worth. The total area under the curve is 1, so if you want the area to the appropriate, then you uncover the area to the left and also subtract from 1. The command also looks like:

(1- ext pnom (280,272,9))

Thus, (P(x>280) approx 0.187)

Therefore 18.7% of all pregnancies last even more than 280 days. This is not unexplained because the probcapacity is better than 5%.

c. First interpret the statement into a mathematical statement.

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P (x

Figure for Example (PageIndex1)c

To uncover the probability on the TI-83/84, looking at the image, though it is hard to watch in this situation, the lower limit is negative infinity. Aacquire, the calculator doesn’t have this on it, put in a really little number, such as (-1 imes 10^99=-1 E 99) on the calculator.