According to a report by the Commerce Department in the fall of 2004,20 % of U.S. households had

According to a report by the Commerce Department in the fall...

Key Concepts

Continuity Correction

Continuity correction is the adjustment made when a discrete distribution (like the binomial) is approximated using a continuous distribution (like the normal). This correction involves adding or subtracting 0.5 to the discrete x-values to improve the approximation's accuracy. It accounts for the fact that a continuous distribution assigns probabilities over intervals rather than specific points.

Z-Score Calculation

Z-score calculation is the process of standardizing a value from a normal distribution by subtracting the mean and dividing by the standard deviation. This standardization allows one to find probabilities associated with the standardized value using the standard normal distribution, which is essential when implementing the normal approximation to compute probabilities for various ranges.

Normal Approximation to the Binomial Distribution

The normal approximation is a method used to estimate the probabilities of a binomial distribution when the number of trials n is large enough for the distribution to be approximately symmetric. This approximation is based on the central limit theorem, which states that the sum of independent random variables tends toward a normal distribution, even if the original variables are not normally distributed. It simplifies calculations by converting the binomial problem into one involving a continuous normal distribution.

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by two parameters: n (the number of trials) and p (the probability of success in each trial). This distribution is commonly used in scenarios where each trial results in a binary outcome, such as success/failure or yes/no.

Transcript

00:01 After analyzing some data from the fall of 2004, it was found that 20 % of u .s.

00:08 Households had some type of high -speed internet.

00:12 So if you envision asking people, do you have internet net? the two answers would be yes or no.

00:18 So the probability of success would be defined as 0 .20.

00:24 In a random sample of 80 people, we want to determine different probability.

00:34 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.

00:45 There's a defined number of trials, and each of those trials would be independent.

00:52 And in part a, we want to determine the probability that exactly 15 of those households had the high -speed internet.

01:05 And we want to use a normal approximation to calculate this.

01:09 So we are going to use the bell shape curve.

01:13 And in order to use that bell shape curve, we need to know an average and we need to know a standard deviation.

01:20 And to calculate our average, we'll multiply n times p.

01:25 So we talked with 80 people or 80 households and the probability of success was 0 .20.

01:35 Therefore there was an average of 16 households that had internet connection.

01:43 To find our standard deviation, we would use the formula of the square root of n times p times the quantity of 1 minus p.

01:52 So again, we looked at 80 households.

01:56 The probability of success was 0 .20.

01:59 And if i do 1 minus 0 .20, i get 0 .80, which would be the probability of finding someone that does not.

02:07 Have the internet connection in 2004.

02:10 And we end up with a square root of 12 .8 as our standard deviation.

02:17 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.

02:34 So when we're using the normal curve, our average, goes in the center, and that would correspond to a z score of zero.

02:47 And we're trying to determine that x equals 15.

02:51 Well, 15 would be to the left of 16 on a number line.

02:55 And binomial distributions are discrete probabilities.

03:01 And when we model discrete probabilities, we model them with histograms.

03:07 So i need you to envision a histogram tower.

03:10 For that number 15.

03:13 And the low end of that tower would be at 15, try it at 14 .5 as a low boundary.

03:23 And the upper end would be at 15 .5.

03:27 So when we're trying to determine the probability of 15, we're trying to determine the probability of this whole tower.

03:35 So therefore, we are going to approximate it by determining the probability that x is between 14 .5 and 15 .5 inclusive.

03:49 Now, in order to use the normal curve, we will have to turn the boundaries of the shaded area into z scores.

03:57 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.

04:08 And to calculate z scores, we use the formula x minus mu over sigma.

04:15 So the z score for 14 .5 could be calculated by saying 14 .5 minus 16, which was the average or the mean, and divided by the square root of 12 .8, which was the standard deviation.

04:31 And that z score calculates out to be about a negative 0 .42.

04:43 Now we want the z score for 15 .5.

04:48 So we'll use 15 .5 minus the mean of 16 divided by the standard deviation, which was the square root of 12 .8.

05:00 And you will get a z score of about negative 0 .14.

05:11 So when we're talking about the area of this curve between 14 .5 and and 15 .5, it's comparable to the probability that z is between negative 0 .42 and negative 0 .14.

05:33 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.

05:44 So when you use the standard normal table, we find the units place and the tenths place along the left side.

05:58 So the tenths place is a four.

06:01 So we're going to have a units place of zero.

06:03 So we're going to be looking up negative 0 .4.

06:06 And then the 100th place was a 2.

06:09 So we're going to look beneath the 0 .02.

06:13 And in doing so, we are going to find.

06:15 An area of 0 .3372.

06:20 So that area is the area from that z score boundary line down into the left tail.

06:33 Now we want to find the z score associated with negative 0 .14.

06:40 So again, we want the units place and the 10th place.

06:44 So we'll look up negative 0 .1.

06:47 The 100th place was a point 0 .04, and where the row and the column meet up in that chart is going to give you an area of 0 .4443.

07:00 So that means from the z score of negative 0 .14 into the left tail is 0 .443.

07:11 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.

07:18 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 we'll subtract those two decimals, 0443 minus 0 .3372 for a probability of 0 .1071.

07:47 So recapping, we have surveyed people and asked them, do they have an internet connection? the probability in 2004 that they had some type of high -speed internet connection was 20%.

08:03 When we surveyed 80 people, the probability that exactly 15 of those 80 had high -speed internet would have been approximated at 0 .1071.

08:16 For part b, we want to determine the probability that at least 20 of those 80 households had the high -speed internet.

08:32 So we're going to approach this in a similar fashion.

08:36 I'm going to draw that bell -shaped curve.

08:40 We're going to put the mean in the center, which again corresponds to that z score of zero.

08:48 20 would be located to the right of that.

08:55 And i want you to envision that histogram tower again...