Earthquakes The magnitude of earthquakes since 1900 that...

Earthquakes The magnitude of earthquakes since 1900 that measure 0.1 or higher on the Richter scale in California is approximately normally distributed, with $\mu=6.2$ and $\sigma=0.5,$ according to data obtained from the U.S. Geological Survey. (a) What is the probability that a randomly selected earthquake in California has a magnitude of 6.0 or higher? (b) What is the probability that a randomly selected earthquake in California has a magnitude less than $6.4 ?$ (c) What is the probability that a randomly selected earthquake in California has a magnitude between 5.8 and $7.1 ?$ (d) The great San Francisco Earthquake of 1906 had a magnitude of $8.25 .$ Is an earthquake of this magnitude unusual in California? (e) What is the percentile rank of a California earthquake that measures 6.8 on the Richter scale? (f) What is the percentile rank of a California earthquake that measures 5.1 on the Richter scale?

Transcript

00:01 All right, for this problem, talking about earthquakes in california, we have that the mean is 6 .2 on the richter scale, and the standard deviation is 0 .5.

00:20 So part a is asking about the probability that an earthquake has a magnitude of 6 .0 or greater.

00:30 So we plug into our normal distribution formula.

00:43 All right.

00:44 The amount we're looking for goes first, 6 .0.

00:48 Then you take out the mean and divide it by the standard deviation.

00:56 So this is negative 0 .2 over 0 .5, which is negative 0 .4.

01:08 All right.

01:08 So we have normal distribution where z is negative 0 .4.

01:15 All right.

01:16 Now, for normal distributions, anything that is greater, you have to subtract one from it because it's always looking for the area under the curve less than the amount that you have.

01:28 So you have to subtract 1.

01:29 All right.

01:30 So if you go to your table of z values and look for negative 0 .4, you'll find that the probability is 0 .3446.

01:45 So the final probability is going to be 0 .6 -554.

02:00 Part b is asking about a magnitude that is less than 6 .4.

02:18 So again, plugging it in, we have 6 .4 minus 6 .2 over 0 .5.

02:26 And this one is just like the last part on part a we have a z value of 0 .4 again and so going to the table you should see that it's the same value 0 .6554 okay for part c it asks about the probability that it has a magnitude between between 5 .8 and 7 .1 all right so same same deal you take the you want to find a z value from the 7 .1 and take that away from the z value you get from 5 .8 so this so this should give you 0 .9 over 0 .5 which is 1 .8 and this one gives you a z value of of negative 0 .8.

04:31 All right.

04:32 Again, going to the tables, 1 .8 has a probability of .9641 and negative 0 .8 has a value of 21.

04:49 21 .19%.

04:53 So subtracting those, you get .752.

05:04 All right.

05:06 Part.

05:07 This wants to know about an earthquake of a magnitude of 8 .25...

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