Question

For each question, give the z-scores and appropriate areas under the curve. You may use Table II, Excel, calculator, or any software to find the areas. Each part is worth 10 points.

1. The diameter of a bolt produced by a manufacturer is a normal random variable with mean 1.20 inches and standard deviation of .005.

a. For a randomly selected bolt, what is the probability that it is less than 1.195 inches in diameter?

Answer 1

I am not for sure how to do this. I missed class because of personal reasons and I'm lost now. If I'm told how to do this I'm sure I can do the rest alone!

Answer 2

Do you know how standard deviation works?

Answer 3

Yes but Im not sure how to find the probability

Answer 4

Also how can we find the Z score I don't think we have enough information. Unless we use 1.195 as x

Answer 5

Well, you can find a lot just from the DS.

Answer 6

Oops. SD...

Answer 7

Z=\[z=(x-\mu)/\]

Answer 8

See, a z score is a measure of how many SDs something is from the mean.

Answer 9

Yes we are given the SD and the mean right? So x=1.195?

Answer 10

If I used that I got -1 as the Z score

Answer 11

"mean 1.20 inches and standard deviation of .005"

So for every .005 it is off from 1.2, it is one SD and therefore has a z of 1 for every SD. The only other question is if it was abive and + or below and -.

So yes, Z is < -1 for anything smaller than 1.195.

Answer 12

Okay that makes sense thanks! How do we find the probability?

Answer 13

You need a z score chart. Is there one in your book?

Answer 14

Yes I have the table

Answer 15

But it doesnt have .005

Answer 16

OK. So, for anything with a z of -1, what do you see?

Answer 17

Do we use .01?

Answer 18

The .005 is already done. That let you find z. Once you have the z you use it to find the probability.

Answer 19

I mean the table you have to use the SD and the z

Answer 20

You already used the SD. The SD got you the Z. Now that you have the Z, -1, you use that on the table.

Answer 21

A Z table tells you what percentage of the area is under the curve from \(-\infty\) up to the Z score you use. The actual calculation can be done with integral calculus, but the table works fone for this.

Answer 22

Thats the table I'm looking at. I see -1 but it gives me many numbers

Answer 23

This one is cleaner:
http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

I can explain it to you.

Answer 24

Well, that is positive. You need negative,,,

Answer 25

Okay I see that. Thanks now don't I have to choose a number up top and follow it down till I get till -1 and that will give me the area under the curve?

Answer 26

Yeah I couldn't find a negative one I have the table in my book with negatives though. Thanks for finding one. I am going to save that.

Answer 27

The first column in the PDF, -3.4, -3.3, -3.2, etc. That is the first few digits.

The top row, .00, .01, .02, is the next digit.

Where the column and row meet is the number you need.

Answer 28

Oh so it would be the .00

Answer 29

Lets say I had -2.15 as a z score. I would go down the first column to -2.1. That is the row I need. Then I would go accross to .05. That is the column I need.

So yes, you want -1.0 and .00

Answer 30

Ah ok I got that so the Area under the curve is .1587?

Answer 31

Is that also the probability?

Answer 32

Yes, which you can take as a percentage too.

Answer 33

c. The bolts are specified to be between 1.193 and 1.21 inches in diameter. What proportion of bolts meet the specifications?
d. What proportion of bolts are within 1 standard deviation of the mean?

Answer 34

Now, have you seen pictures of the normal curve? Like:
http://www.afb.org/images/celeb_sol-figure2.gif

Answer 35

How would we do those? Yes I have thanks!

Answer 36

Hehe. The answer to c is right there...

Answer 37

oops. I mean d.

Answer 38

Well I mean how would we solve it mathematically?

Answer 39

Would that be half of 68%?

Answer 40

OK. From a Z chart?

Answer 41

Or 68%?

Answer 42

Within 1 is \(\pm 1\) so no, you do not cut it in half.

Answer 43

Yes from a Z chart thank you!

Answer 44

Or more accuratly, 68.26%, which I can show you how to get from that chart.

OK. Now, you saw that below -1 was .1587, right? What about postive 1 Z? What number is that?

Answer 45

Ok that would be nice thanks!

Answer 46

.8413

Answer 47

OK. So, what do these numbers really mean? Well... let me draw this if I can...