Find the following z values for the standard normal variable...

Find the following z values for the standard normal variable Z. (You may find it useful to reference the z table. Negative values should be indicated by a minus sign. Round your answers to 3 decimal places.) a. P(Z ≤ z) = 0.1054 b. P(z ≤ Z ≤ 0) = 0.1344 c. P(Z > z ) = 0.7146 d. P(0.26 ≤ Z ≤ z) = 0.3123

Transcript

00:01 We want to find some z scores.

00:03 So i'm going to start by drawing the standard normal distribution, which gives us the probability distribution for z.

00:12 So its mean, mu is 1, its standard deviation is, its mean mu is 0, its standard deviation is 1, but we don't really need that here.

00:23 Okay, we have been told the probability of getting something below our little z is 0 .1054.

00:31 So the total area under this curve is 1, it's a probability curve, and it's symmetric.

00:37 So below the mean is 0 .5, above the mean, 0 .5.

00:41 So for part a, z is down here somewhere.

00:45 So that this area is 0 .1054.

00:51 Now we need to find z.

00:54 Okay, so we know the area, we need to turn it into z.

00:58 This is done with the inverse normal function.

01:02 If you're using excel, it's called q norm, if you're using r, your calculator should have this as well.

01:08 It is a cumulative function.

01:10 So you put in the area to the left, and it gives you z.

01:15 This is the easiest way to do this, but maybe you want to use a standard table.

01:20 It has to be a table that has at least three decimal places on the z score, because we want our answers to three decimal places.

01:27 And it would look a little bit like this.

01:30 Z scores of rows and columns, probabilities in the middle.

01:34 But you're not looking for 0 .1054.

01:37 The standard is interested in the area between z and 0.

01:44 So you would be looking for 0 .3946 and matching it up to z.

01:50 The standard also won't tell you if your z score is positive or negative.

01:55 It doesn't know.

01:56 If you imagine a score over here, where this bit is 0 .1054, you put in the same area, it can't tell which one you're looking at.

02:06 So if your score is below the mean, it's negative, if it's above the mean, it's positive.

02:11 So this one is going to be negative.

02:13 I'm going to use my cumulative function here.

02:18 1054 becomes minus 1 .251.

02:29 Hoppy, it's over here.

02:41 So now the area between, so area between z and zero is 0 .1344.

02:54 So this bit is 0 .1344.

03:05 Okay, so if you were using the table, you're looking for that, and your score is going to be negative again because it's below the mean, because z is on the left side of this interval here.

03:16 If you're using the cumulative function, we need to put in this area.

03:21 And that would be 0 .3656 .6 .6...

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