Suppose that 64 % of stocks recommended by a broker increase...

Suppose that $64 \%$ of stocks recommended by a broker increase in value within six months. Use the normal approximation to the binomial distribution to determine the probability that of 372 recommendations, (a) at least 225 will increase in value and (b) more than 250 will increase in value.

Key Concepts

Binomial Distribution

The binomial distribution represents the number of successes in a fixed number of independent experiments (or trials), with each experiment having two possible outcomes (success or failure) and a constant probability of success. It is used to model events where there are a fixed number of trials and each trial is independent from the others.

Normal Approximation to the Binomial Distribution

The normal approximation to the binomial distribution is a method used to approximate binomial probabilities using the normal distribution when the number of trials is large. This approximation is valid when both np and n(1-p) are sufficiently large, allowing the discrete binomial distribution to be treated as a continuous one.

Mean and Variance of a Binomial Distribution

In a binomial distribution, the mean (expected value) is calculated as np and the variance is calculated as np(1-p), where n is the number of trials and p is the probability of success. These parameters are essential for characterizing the distribution and are used in the normal approximation.

Continuity Correction

The continuity correction is a technique used when approximating a discrete distribution (such as the binomial) with a continuous distribution (such as the normal). It involves adjusting the discrete value by 0.5 to better approximate the probability under the continuous curve, leading to more accurate results.

Transcript

00:01 So we have a binomial setting with the proportion being .4 and n being 25.

00:07 So the x value, the number of successes, could be anywhere from 0 up to 25 successes.

00:14 And we're going to actually answer part b first.

00:17 Part b asks you to find the mean, which will be n times p, which will be that .4 times 25, and that's going to give us 10.

00:26 And then the standard deviation of x is going to be the square root of 25 times .4 times .6, and then we'll square root that answer.

00:37 And when we do that, we get that value to be approximately 2 .449.

00:45 And i'm just going to store that in my calculator as x, actually.

00:49 Now we'll go back to a.

00:51 We want to look and see is the mean plus or minus three standard deviations, does that include this interval? and so we take that mean plus or minus three times that 2 .449, and 10 minus 3 times 2 .449 gives us about 2 .652.

01:17 Changing that to addition, that is about 17 .35.

01:25 And yes, this would then be approximately normal, because that interval does include these x values.

01:31 So it's appropriate.

01:33 So now let's go to part c.

01:34 Part c, we want to find the exact value for the probability that x is greater than or equal to 9, which we're going to do via complement.

01:42 We're going to find 1 minus the likelihood of x being less than or equal to 8...

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