After analyzing some data from the fall of 2004, It was found that 20% of US households had some type of high speed internet. So if you envision asking people do you have Internet net to answer would be yes or no. So the probability of success would be defined as .20. In A random sample of 80 people, we want to determine different probabilities. Now this information is a binomial probability because there are only two outcomes, yes or no. Either they have internet connection or they don't. There is a defined number of trials and each of those trials would be independent and in part a we want to determine the probability that exactly 15 of those households had the high speed internet and we want to use a normal approximation to calculate this. So we are going to use the bell shaped curve. And in order to use that bell shaped curve, we need to know an average and we need to know a standard deviation and to calculate our average will multiply N times p. So we talked with 80 people or 80 households And the probability of success was .20. Therefore There was an average of 16 households that had Internet connection to find our standard deviation, we would use the formula the square root of n times p Times The Quantity of one -P. So again, we looked at 80 households, the probability of success was 0.20 and if I do one minus 10.20, I get 0.80, which would be the probability of finding someone that does not have the internet connection in 2000 and four and we end up with a Square root of 12 8 as our standard deviation. Now we could use our calculator and get a decimal approximate for that. But we highly recommend that you use the exact value throughout all our calculations, rather than a decimal approximate to get the most accurate probability. So when we're using the normal curve, our average goes in the center And that would correspond to a Z score of zero and we're trying to determine That x equals 15 or 15 would be to the left of 16 on a number line and by no meal distributions are discreet probabilities and when we model discreet probabilities, we model them with hissed a grams. So I need you to envision a hissed a gram tower For that number 15. And The low end of that tower would be at 15. Try at 14.5 as a low boundary And the upper end would be at 155. So when we're trying to determine the probability of 15, we're trying to determine the probability of this whole tower. So therefore we are going to approximate it by determining the probability that X is between 14.5 And 15 5 inclusive. Now, in order to use the normal curve, we will have to turn the boundaries of the shaded area into Z scores. So we're going to need a Z score associated with 14.5 and we're going to need a Z score associated with 15.5. And to calculate Z scores, we use the formula x minus mu over sigma. So the Z score for 14.5 could be calculated by saying 14.5 -16, which was the average or the mean and divided by the square root of 12.8, which was the standard deviation. And that Z score calculates out to be about a negative .42. Now we want the Z score for 15.5, So we'll use 15.5 minus the mean of 16 divided by the standard deviation, which was the square root of 12.8. And you will get a Z score of about negative .14. So when we're talking about the area of this curve Between 14.5 and 15.5, It's comparable to the probability that Z is between negative .42 And -14. So at that point we will need to use the table, the standard normal table in the back of your textbook to find the area that's associated with each of these Z scores. So when you use the standard normal table, we Find the units place and the 10th place along the left side. So the tense places of four. So we're going to have a units place of zero. So we're gonna be looking at negative 04 And then the hundreds place was a two. So we're going to look beneath the .02. And in doing so we are going to find an area of .3372. So that area is the area from that Z score boundary line down into the left tail. Now we want to find the Z score associated with negative 0.14. So again we want the units place and the tense place. So we'll look up negative 0.1. The hundreds place was a point oh four. And where the row and the column meet up in that chart is going to give you an area of .4443. So that means from the Z score of negative 14 into the left tail Is .4443. And we're looking for the area in between. So in order for us to find that difference we're going to have to subtract. So we're going to have to subtract the probability that Z was less than negative 0.14 minus the probability that Z was less than negative 0.42. Or will subtract those two decimals? 20.4443 minus 0.3372. For a probability of .1071. So recapping, we have surveyed people and ask them do they have an internet connection? The probability in 2004 that they had some type of high speed interconnect Internet connection was 20% When we surveyed 80 people, The probability that exactly 15 of those 80 had high speed internet Would have been approximated at .1071 for part B. We want to determine the probability that at least 20 Of those 80 households had the high speed internet. So we're going to approach this in a similar fashion. I'm going to draw that bell shaped curve. We're going to put the mean in the center, which again corresponds to that Z score of zero 20 would be located to the right of that. And I want you to envision that hist a gram tower. Again, The low class boundary would be 19.5. The upper boundary of that tower would be 25. And we're trying to determine the probability that were greater than 20 Were equal to 20. So therefore are shaded region really begins at 19.5. So we're going to approximate this by determining the probability that X is greater than or equal to 19.5. Therefore, we need the Z score associated with that 195. So we'll use that Z score formula X minus mu over sigma. And you will get a Z score of approximately .98. So when we refer to X being greater than or equal to 19 5, It's comparable to the probability that Z is greater than .98. So we will have to go back to that table in the back of the book. Again, you're standard normal table. And we look up the units place in the tents, place down the left side and the hundreds place across the top. And in doing so you are going to get an area of 8365. So that means From here into the left tail is .8365. But we want to go into the right tail. So instead we're going to do one minus the probability that Z was less than .98 or one minus that decimal of 8365 For an overall area of .1635. So the probability that At least 20 households had high speed internet access Would be approximated at .1635 in part C, you were trying to determine the probability that fewer than 10 households had that high speed internet access. So here's our bell curve again with mean in the center We're going to envision 10 to the left of that. We need to envision the tower. So if we're going to be less than that, we need to envision the next tower over as well which would have been the tower for nine. So when we want less than 10, we're really saying that we want the probability that x is less than or equal to nine. So when we focus on the tower for nine, the low boundary of that tower would be at 8.5. The upper boundary would be at 95. And we are trying to Find the area less than nine Were equal to nine, Placing our boundary of the shaded region at 9.5. So we'll have to use the probability that X is less than or equal to 9.5 to approximate this probability We will need the Z score for 95. Z score for 95 is calculated using x minus mu over sigma. And you're going to get an approximate value of -1, So therefore, the probability that X is less than or equal to 9.5 can be compared to the probability that Z is less than negative 1.82. So we'll go to our standard normal table. We'll find the Units place and the 10th place along the side. We'll find the hundreds place across the top And that yields an area of .0344. Going into the left tail. So from Zeon into the left tail is an area of .0344. So therefore our probability Is going to be .0344. So the probability that fewer than 10 Households had high speed Internet access can be approximated at .0344. And to conclude this problem, let's look at part D. In part D. We want the probability that between 12 and 18 households inclusive, had that high speed internet access. So we'll take the same approach. We'll place the mean in the center, which corresponds to that Z score of zero Will place 12 to the left of that 16 And 18 to the right. We will envision the history Graham towers and their boundaries. The low boundary of the tower for 12 would be 115. The upper boundary would be 12.5 With the tower for 18. The lower boundary would be 17 5 And the upper boundary would be 185. And we are trying to determine the probability that X is between 18 and 12 or equal to. So therefore are shaded areas are defined By the 11.5 and the 18.5. So to approximate this, we're going to determine the probability that X is between 11.5 And 18.5. Thus requiring Z scores for 11 5, An 18.5. So the z score for 11 5 will be calculated To be approximately a negative 1- six. And the 18 5 when you apply, your formula Will be approximately a 70. So therefore, if we were determining the probability that X is between 11 5 and 18 5, It's comparable to the probability that Z is between negative 1-6 and Positive .70. We will have to rely on our chart again in the back of the book, we're going to find our units and tense place and then we'll find our 100th place. So we get a corresponding area To be .1038. So that's saying from AZ of negative 1.26 into the left tail has an area of 1038. For the Z score of 70 we're going to look up the units place in the 10th place so we'll look up 07 The hundreds place was zero And you will find a corresponding area of .7580. So from the Z score of 70 into the left tail Is .7580. So if I want the area in between, I'm gonna have to look for that difference. So I'm going to need the probability that Z was less than .70 minus the probability that Z was less than -1- six. So I'm gonna take those two decimals 7580 -2038. To get an approximate probability of 6542. So the probability when selecting 80 households that between 12 and 18 will have high speed Internet access back in 2004 Was approximated at about 6542. And that concludes the four parts to this problem