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3) Answer the following questions using either your graphing calculator (show calculator commands) or the Table of Critical Values for the Standard Normal Distribut...

Question

3) Answer the following questions using either your graphing calculator (show calculator commands) or the Table of Critical Values for the Standard Normal Distribution (Z- Table) Find the proportion of observations from a standard normal distribution that falls in each of the following regions_ In each case, shade the area representing the region:a. 2 <-1.75b. 22-2Your answer:Your answer:2 > 1.5d.-<z<1.5Your answer:Your answer:

3) Answer the following questions using either your graphing calculator (show calculator commands) or the Table of Critical Values for the Standard Normal Distribution (Z- Table) Find the proportion of observations from a standard normal distribution that falls in each of the following regions_ In each case, shade the area representing the region: a. 2 <-1.75 b. 22-2 Your answer: Your answer: 2 > 1.5 d.-<z<1.5 Your answer: Your answer:



Answers

For Exercises 47 to 50, use Table A to find the proportion of observations from the standard Normal distribution that satisfies each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to the question. Use your calculator or the Normal Curve applet to check your answers.

Table A practice
$\begin{array}{ll}{\text { (a) } z< -2.46} & {\text { (c) } 0.89< z <2.46} \\ {\text { (b) } z >2.46} & {\text { (d) }-2.95< z <-1.27}\end{array}$

This question involves the normal curve and your standard normal probabilities table. So the first question we want to know, what's the probability of Getting a value that's got a Z score less than 2.85. So on my bell curve I'm going to go to 2.85 which would fall right about here. And I want to know less than that. So the probability over here or the area underneath the bell curve to the left of 2.85. So we're gonna go to our table. Yeah. And look up a value of 2.8 Along the left column and .05 on the top. And where those two rows and columns intersect is going to be our probability .9978. So in other words to left of 2.85 as our Z score 99.78% of the data falls within their. Now the second question is, what's the probability greater than 2.85? So that would be this chunk right here at the tail end. I'm gonna color that in in green. And for that we just have to realize the entire bell curve has an area of one. So if we do one minus that 10.9978 that we just had is the answer for part A. That's gonna give us .00-2 for A probability Z score greater than 2.85. Now the next question is Z greater than negative 1.66. So I'm gonna come here negative 1.66. Let me raise this .9978 here. So in blue negative 1.66 falls right about there and we want to know this probability everything greater than that. So I'm gonna look up negative 1.6 And then .06 in the top of my standard normal probability table and where those intersect. That will be my probability .0485. But you should notice in the little picture at the top, the probability inside of that table is always to the left of your your Z score marker. So that .045 is actually this part of the bell curve to the left of -166. So to the right of -1.66 is gonna be 1 -1485 Which is .9515. So all of this area to the right Of -1.66 is .9515. and now the last question wants just the area from negative 1.66. Up until 2.85. Well for this one, all we have to do is take our .9515 answer, which was all the way to the right, and subtract that little tiny .0022 part at the at the tail end there. So .9515 -1022 gives us .9493 as our probability Of AZ score, following between negative 1.66 and 2.85.

Okay for this question we're gonna be using your bell curve and your standard normal probability table Table A in the back of your book. So first one we want to find out what's the probability of getting a Z. score between negative 1.33 and 1.65. So I'm gonna use this bell curve on the left here. And I'm gonna mark on here where negative 1.33 is And where 1.65 is right about here. And essentially we want to figure out the area underneath the bell curve between those two values. So this space in here. Now to figure out that area or the probability of landing in this section, we're going to need The table in the back of the book. So I'll start with the 1.65. We're gonna look up 16 in the left hand farthest left hand column. And then you're gonna look up .05 at the top role. And where that column and row intersect Is the probability of less than 1.65 which is .9505. So from 1.65 to the left, The area is .9505. But as you look at 1.65 to the left, we don't want all the way to the left. We want to stop at negative 1.33. So we need to find the probability of landing less than negative 1.33. So we'll look up negative 1.3 in the left hand column and .03 in the top rose. And they intersect at .0918. And then to find the probability between negative 1.33 and 1.65. We're gonna subtract our two probabilities 20.9505 minus point oh 918 Gives us a probability of .8587. Now we'll do the same thing but we're gonna switch. Z scores 2.5 and 1.79. So on my bell curve on the right I'm gonna mark .5 and .1.79 right about here. And we want this area In between those two values. So I'll start with the 1.79. The probability of getting a Value less than 179 as the Z score. So look up 1.7 and then .09. They intersect at .9633. So to the left of 1.79 is .9633. Mhm. Home. And now we'll do the same thing for 05050 Is .6915. So the probability less than 0.5 point 6915. And then to find our area in the little section between 1.5 and 1.79. We're gonna subtract those two probabilities 20.9633 minus 0.6915 Is .2718. Yeah 0.2 718 is the probability landing between .5 and one point so.

Yeah we're gonna be using our normal distribution along with your standard normal probability table A. K. A. Table a in the back of your book to figure out the percent of the time R. Z score should land between negative 2.5 and 0.78 for the first question. So let's do the 0.78 1st. In your standard normal probability table you're gonna look up 0.7 on the left hand column and then 8.8 at the very top row. And where that column and row intersect is going to be your probability less than 0.78 Which happens to be 0.78 to three. So I'm just gonna make a little marker here up so changed to red to the left of A. Z. Score a 0.78 point 7823 Now notice though we don't want all the way to the left. We just want all the way to negative 2.5 So we need to get rid of the little probability the little part of the bell curve that is less than a Z score of 2.5 So we'll do the same thing. Well look at negative 0.20 and then 0.5 at the top of your chart. And where that row and column intersect is gonna be our probability 0.202 So to find our probability that's going to go from negative 25 2.78 We're gonna do 0.78 to three. Yeah minus 0.202 And that gives us 0.76 to one. And I'll read that right in here as our answer. 0.7621 76.2% sent chance of landing between negative two point oh five and 50.78 Now the second part of the question, we're going to do the same process but with different Z scores negative 1.112 negative 0.32 So let's start with negative 0.32 So negative 0.3 at the left hand side of the table and then 0.2 at the very top. Those intersect at 0.3745 So to the left of negative 0.32 is a probability of 0.3745 But again, we don't want to go all the way to the left. We just want to go to negative 1.11 So we need to figure out the probability in this little section to the left of negative 1.11 So back to our table negative 1.1 on the left hand side of the table and then point no one at the top those that column and row will intersect at 0.1335 So this probability right in here is 0.1335 So to find just this green colored section, we're going to do 0.3745 minus 0.1335 which will give us a probability of point 241 So a 24.1% chance of landing at a Z score between negative 1.11 and negative 0.32

So we're trying to use the standard normal table, which in this particular one is table A To determine the 10th Z score that goes with the 10th%ile. So let's start by drawing a picture. So here is our normal curve And this is part a. The 10th%ile means that we're looking for the location where 10% of the curve lies below that value. So we're looking for this Z score. So we're supposed to start by using the table. So what you're looking for is in your Z table in your standard normal table, you're looking for the closest number in the belly of the table to be .1000. And the reason being is .10 or 10%ile Out to four decimal places would have two more Zeros. Now you're not going to find that. So what we need to do is we need to go to the closest number and the closest number two that is going to be .1003. So we want to work our way back out of the table And when you go back out of the table to the column on the left, you're going to find the number negative 1.2. And when you follow up, you're going to find the .08. And when we put them together, that's telling us that the Z score that separates the bottom 10% From the Upper 90% would be a negative 1.28. Now we're supposed to check our work utilizing a calculator or an app or an applet. So I'm going to bring in my calculator and I have the texas instruments calculator and in order to do this, I'm going to use the inverse norm feature on the calculator. And when you use inverse norm you have to provide three pieces of information. We have to provide the area in the left tail, the mean and the standard deviation. So for our problem, The area in the left tail is .10. Since we are working with Z scores the mean of the standard normal table is always zero and the standard deviation is one. So to access inverse norm on the texas instruments calculators you're gonna hit second and the bears button. Sorry, try again second and the bears button and it happens to be number three in my menu. So we're looking for inverse norm. We're going to type in the area in the left tail, followed by the mean and the standard deviation. And sure enough, we get an answer close to negative 1.28. Now the calculator is going to be a little bit more accurate. Then the table, the table is only going out to two decimal places. And remember when we did this, we went close 2.1000, it wasn't perfect. So for part B we are looking for the Z score That separates the top 34%. So we want 34% greater than that Z score. So because our standard normal tables, usually at the top, you're going to see a little image and in that image you're going to see that it's shaded to the left, which is saying that to use this table, we've got to talk about the area in the left tail. Well, our picture here doesn't have anything in the left tail. So we're going to have to put something into the left tail and keep in mind that the left side plus the right side always has to add up to one. So a .66 plus .34 is going to have the total area underneath that bell shaped curve to be one. So, again, this time we're looking for .66 in the table, we're looking for the closest number two that we can always put two zeros on it. And if you scour that table, the closest number 2.6600 is going to be .6591. Again, we're gonna work our way back out of the chart. When you go to the left, you're going to find 0.4 and when you go up You're going to find .01. So that means the Z score, that is separating the top 34% of the curve from the bottom 66% of the curve is going to be point 41, and again we can use inverse norm as a built in check. So if you use inverse norm again, you're going to provide the area in the left tail, followed by the mean and the standard deviation of the standard normal curve. Again, I'm going to bring in my calculator And I'm going to hit 2nd. There's I'm gonna select inverse norm. The area is 66 Standard, meaning zero standard deviation is one. So we're getting a Z score of .14, And sure enough, it's close to what we got out of the table. So that is how you use the table to find the Z score associated with any given area in the left tail.


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