In 14-17 , use the normal approximation to estimate each probability. Round your answers to thre

In 14-17 , use the normal approximation to estimate each...

Key Concepts

Standardization

Standardization involves converting a random variable to a standard normal variable (z-score) by subtracting the mean and dividing by the standard deviation. This process allows the use of the standard normal distribution to find probabilities and is a critical step when applying the normal approximation to assess probabilities.

Mean and Variance Calculation

For a binomial distribution, the mean and variance are central to the approximation process. The mean is computed as n multiplied by p, and the variance is calculated as n times p times (1-p). These values are used as the parameters of the approximating normal distribution.

Binomial Distribution

This concept refers to the probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. It provides the basis for calculating the probability of obtaining a given number of successes in experiments characterized by binary outcomes, such as success or failure.

Normal Approximation

Normal approximation is a method used to estimate the probabilities of a binomial distribution using a continuous normal distribution when the number of trials is sufficiently large. This approach simplifies calculations by allowing one to use the characteristics of the normal curve to estimate binomial probabilities.

Continuity Correction

When approximating a discrete distribution such as the binomial with a continuous distribution like the normal, a continuity correction is applied. This adjustment compensates for the discrepancy between discrete outcomes and continuous intervals by typically adding or subtracting 0.5 to the discrete variable, thereby improving the approximation accuracy.

Transcript

00:01 When it's all said and done, what this problem is really about is you being able to plug in a low, a high, a mean, and a standard deviation into a calculator for it to perform the normal cdf function.

00:14 All right.

00:14 So if you have a calculator that's got a normal cdf function, that's great.

00:18 If not, there are calculators available online.

00:22 If you just search for an online calculator with normal cdf function, you are supposed to be using a calculator at the end of this.

00:28 Now before we can do that though, we have to actually figure out what our low is, what our high is, what our mean is, and what our standard deviation is.

00:36 We have to figure out what all of them are.

00:38 That's the part that's actually the work part of this problem.

00:43 Now to begin with, we are being asked what the probability is that we would have more than 20 successes.

00:48 We are told that the number of trials that we are doing is 200, and that the probability of success is one -tenth, one out of 10.

00:58 Tries, one in ten tries.

01:00 Okay? well, if we are asked what the probability is that we would have more than 20 successes.

01:09 Well, what is the first way that we could have that happen? well, the first way we could have that happen is we could have 21 successes, right? it does say more than.

01:19 It doesn't say 20 or more.

01:21 It says more than 20 successes, right? so the first time, the lowest amount of successes that we could have and still be okay with this problem would be 21.

01:31 So that would be our low to begin with for this problem.

01:35 That is the lowest amount of successes that we can possibly have for this to be, to be answering this problem.

01:42 Our high would then be how many successes can we possibly get.

01:48 If every trial was a success, what is the most amount of successes we can have? well, n is 200.

01:54 N represents the total number of trials.

01:57 If we're only doing 200 trials, it'd be kind of hard to get more than 200 successes...

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