Assume that x is a binomial random variable with n=120 and p=.50 . Use a normal approximation to

step 1 Hello, welcome to this lesson. In this lesson, we have extra this binomial random variable with

Assume that x is a binomial random variable with n=120 and...

Key Concepts

Standardization (z-scores)

Standardization involves converting a normally distributed variable to a standard normal variable (z-score) by subtracting its mean and dividing by its standard deviation. This process makes it possible to use standard normal tables or software to find probabilities.

Continuity Correction

The continuity correction is used when approximating a discrete distribution, like the binomial, by a continuous one, like the normal. It adjusts for the fact that the binomial distribution only takes integer values, generally by adding or subtracting 0.5 in probability calculations.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. It is key to many probability and statistics problems involving binary or yes/no outcomes.

Normal Approximation to the Binomial

When the number of trials is large and the probability of success is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution. This simplifies calculations since the normal distribution is well-tabulated and easier to work with for computing probabilities.

Transcript

00:01 Hello, welcome to this lesson.

00:03 In this lesson, we have extra this binomial random variable with simple size 120 and the probability of success at 0 .5.

00:14 So here we use the normal apposumation to find the probability that x is less than or equal to 55.

00:25 So because we had the original values, the original distribution in binomial, we'd have to convert.

00:35 To the normal distribution by finding the mean, which will be called to the n times the p.

00:51 So we have the n as 120 times the p which is 0 .5.

00:57 This is equals to 60.

00:59 So we have the mean of the distribution as 60.

01:05 Then we also need to find a standard deviation here.

01:12 The standard deviation is equals the square root of the n times p which is roughly 1 minus p so here would have x the standard deviation which is equals to n times p is already 60 but that would put everything in there so 0 .5 times 1 minus 0 .5 which is 0 .5 again so x will be equal to 120 times 0 .5 5 times 0 .5.

01:57 It means square root of that.

01:58 So we have square root of 30.

02:00 Then to the small places, we have 5 .4 .8.

02:08 So this is the standard dimension and the mean.

02:13 Okay, so answering the first question, we'll be looking at probability that x is less than or equal to 55.

02:22 We'll need to use the z's score to find this so here would find a z score of 55 so this is equals to the x minus the mean all over the standard deviations he would have 55 minus 60 all over 5 .48 and this keeps us 5 divided by negative 5 divided by okay so here so here the probability that z is less than or equal to negative 0 .91 which is equals to the probability that x is less than or equal to 55 is equals to 0 .1814 so that's the probability now that's the first part let's look at the second part that certain cal we are looking at the range so we have the probability that x is between 60 and 80 okay so here we know that the mean is 60 so the probability that the x is 60 would be equal to zero uh there z score for that will be zero which will translate to a probability that x z is less than all equal to zero as 50%.

05:07 Okay, because the mean is zero.

05:16 The mean for this one is 60.

05:20 So the probability that x would be equal to, or it would be greater than this is where i actually do.

05:31 So probability that x would be less than to 60 would be equal to 50%.

05:48 Okay, it would be this.

05:51 So this would be the mean, right? the whole of this would be equal to 50%.

06:01 All right.

06:04 So with this out of the way, let's look at the probability that x would be equal to would be less than or equal to 80 so that we can subtract the two so looking at probability that x will be less than equal to 80 this is how we do it we find a z score for this so we have the x minus the mean all over the standard deviation so here we have 80 minus 60 all over 5 .48 so the very reason why the z score for the mean for the mean would be 0 is this would have been 60 minus 60 okay so this is equals to you 20 divided by 5 .4 each and that is 3 .6 5 so looking at this finding the probability that is less than or equal to 3 .65 which is equivalently probability that x is less than equal to 80...

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