[szamy004]

A new muscle relaxant is available. Researchers from the firm developing the relaxant have done studies that indicate that the time lapse between administration of the drug and beginning effects of the drug is normally distributed, with mean of 38 minutes and standard deviation of 5 minutes.
a. The drug is administered to one patient selected at random. What is the probability that the time it takes to go into effect is less than 35 minutes?
b. The drug is administered to a random sample of 10 patients. What is the probability that the average time before it is effective for all 10 patients is less than 35 minutes?

[szamy004]

Please help me :)

[morgaine300]

Adding the "please help me" isn't getting you anywhere. We kind of already figured you wanted help. You have to wait for someone to get here, and decide they want to post.

You need to learn how to do these once and for all and not keep posting problem after problem. Learn how to do one first, and then try to apply it to more of them on your own. And these are darn difficult to explain in a forum like this cause it's easier if you can draw the distribution and show how everything relates.

Does your book have problems that ask you to find the probability of just z's? That is, the probability of z > -1, z < 2.5, z < -2.3, stuff like that? If so, you might want to go practice those some more using the chart. It helps if you get a feel for that part before applying to applications.

The z is just how many standard deviations. I like to draw a chart for the x's, and then put a z line underneath it, lined up with the standard deviations in the same place. In the example at the bottom, my mean is 100 and standard deviation is 15. So I've plotted those out at the appropriate spots at 1, 2 & 3 deviations both right and left of the mean. (i.e. 100 + 15 = 115, so that's the 1st standard deviation, etc.) Then underneath I draw z. That shows how many standard deviations. Notice the 115 lines up with 1 on the z line. Because that's at 1 standard deviation. 130 lines up with 2 cause that's at 2 standard deviations, etc. 85 lines up with -1 cause that's 1 standard deviation to the left.

But what if it's not at an even standard deviation? What if I want to know the equivalent z score at 123? Well, 123 is 23 away from the mean of 100. ( or -- i.e. that point minus the mean. That's the distance.

Now, how many standard deviations away is that? Well, a standard deviation is 15. So how many 15's? 23/15 = 1.53. There's 1.53 15's. Hence 1.53 standard deviations. I've shown this spot on the example.

So we took the distance from the mean and divided by the standard deviation, which tells us how many standard deviations away it is. Hence, the z-score equation:





So z = 1.53. I take those to 2 decimal places because that's general what's on the z charts. If we know the z score, that is, how many standard deviations away from the mean our x is, then we can use just one chart based on what the bottom line looks like.

That's part 1. Next one I'll shade an area and we'll do a probability. (If I get to it tonight.)

[szamy004]

Thanks for explaining, but that's the thing I have been trying but I'm only given lecture notes which arnt helpful at all. It's a semester course. So they don't provide a text book for it. So all I have to use is my lecture notes. The reason I post so many is because I do what I can then post here to see if it's right. And you guys explain it better then what I am taught.
Thanks for your help though. I'm doing it now and it's slowly sinking in
:-)

[morgaine300]

OK, now we want to know what the probability is that our x is less than 123. In applications, that x of course will stand for something. So I've made this IQ scores. Supposedly the mean is 100, by definition. I sort of made up the 15 - I had a problem with IQ scores once and I think it was around that. So the problem wants to know the probability that some randomly selected person has an IQ of less than 123.



As a side note, it wouldn't matter if it included "equal" or not. That is, "less than" and "less than or equal to" are the same thing.

The green area is the area we are looking at, cause that's the "less than" part. We already know that's the equivalent of 1.53 in z's. So in a sense, it's like asking what's the probability that z is less than 1.53?

The pictures work nicely. I always draw them because I'm very visual. It helps a lot when you're learning them. So less than is to the left and includes all that green area. Something important to note is the dark purple line at the mean, the halfway point. What do all probabilities have to add up to? One. That means everything under that curve equals one. And what's half that? 5. So everything to the left of the purple line is .5 and everything to the right of that purple line is .5. (Note that I'm talking about the area now, not the line itself. Line is x's or z's. Probabilities is the area.)

So notice that our area includes the entire left side, PLUS a portion of the right side. So we know our probability is greater than .5. Now the tricky part is explaining how to get the probability off the z chart, because there charts can be different. So here's where you have to learn how to interpret the chart. (Aren't you the one I gave the link to of different ones so you could pick which one looks like yours?)

If your chart looks like my example, that is, with over half of it shaded in, then you've got it made, cause all you have to do is look up 1.53 and you've got your answer. If, however, you've got the kind that shades from the center to some point to the side, that's only showing from the halfway mark. That means you'd have to add .5 to it so that it also includes the entire left side.

The probability is .937. Check out your chart and see if you can figure out how to come up with that.

I'd show another example for the probability of x being greater than the 123, but I'm getting kind of sleepy now.

But the concept works the same for all of them. Draw the chart. Put the mean in the middle. You can put the numbers at each standard deviation if you like - it can help give an idea about where your x is located. (For instance, I knew my z was 1.something cause of where it's located.) The z line underneath is just for the visual. Use the z-score equation to get the z for any given x the problem gives you. Then shade the area you want. Less than is to the left, more than is to the right. Remember that "no more than" means to the left. Then go to the z chart to find the probability.

This is for a population. There's a slight twist on it when doing sample means, but it's not a huge deal. But that would take some more explanation... the concept is exactly the same. So if you can figure this out, you'll be able to figure that out.

[morgaine300]

Thanks for explaining, but that's the thing I have been trying but I'm only given lecture notes which arnt helpful at all. It's a semester course. So they don't provide a text book for it. So all I have to use is my lecture notes. The reason I post so many is becuase I do what I can then post here to see if it's right. And you guys explain it better then what I am taught.
Thanks for your help though. I'm doing it now and it's slowly sinking in
:-)

Then post whatever you've come up with so someone can see if it's correct, instead of just the problem.

[morgaine300]

If you can use whatever chart you want, I suggest this one as the easiest when it's new:
http://business.statistics.sweb.cz/normal01.jpg

And I'm going to bed now. :-)

[chaddy]

five hundred ball bearings have a mean weight of 5.02 grams(g) and a standard deviation of 0.30 g. Find the probability that a random sample of 100 ball bearings chosen from this group will have a combined weight of (a) between 496 and 500g and (b) more than 510g