Suppose that X is a binomial random variable with n=200 and p=0.4 (a) Approximate the probabilit

Suppose that X is a binomial random variable with n=200 and...

Key Concepts

Approximating Exact Probabilities

When interested in approximating probabilities for specific outcomes in a binomial setting using the normal distribution, one must account for the fact that the normal distribution is continuous. For exact outcomes, such as finding the probability of a precise value, the approximation involves calculating the probability within a narrow interval around that value, often using the probability density function of the normal distribution along with the continuity correction.

Standardization (Z-scores)

Standardization is the process of converting a normal random variable to a standard normal variable with a mean of 0 and a standard deviation of 1. This is achieved by subtracting the mean and dividing by the standard deviation. The resulting z-score can be used to determine probabilities from standard normal distribution tables or computational tools.

Continuity Correction

When approximating a discrete distribution, like the binomial, with a continuous distribution, such as the normal, a continuity correction is applied. This correction involves adjusting the discrete variable by 0.5 to account for the difference in nature between discrete and continuous variables, thereby providing more accurate approximations of probabilities.

Normal Approximation to the Binomial

For a binomial distribution with large n and where both np and n(1-p) are sufficiently large, one can approximate the binomial probabilities using the normal distribution. This approximation simplifies calculations, especially for probabilities that involve ranges of outcomes, and is based on the Central Limit Theorem which states that the distribution of the sum (or average) of a large number of independent random variables tends to be normal.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli 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 fundamental in discrete probability theory and is widely used to model binary outcomes in various contexts.

Transcript

00:01 Here we have x, a binomial random variable with n equals 200 and p equals 0 .4.

00:09 And we are asked to approximate some probabilities for x.

00:13 Now we can use the normal approximation to the binomial if n times p is greater than 5.

00:23 And if n times 1 minus p is also greater than 5.

00:30 So here we have n times p is equal to 80, and n times 1 minus p is equal to 120.

00:48 Both are significantly bigger than 5, which suggests that we can use the normal approximation as a good approximation.

01:05 For a, we are looking for the probability that x is less than or equal to 70.

01:16 And since we're going to use the normal approximation, we want to include the continuity correction factor.

01:27 That is looking for the probability that x is less than or equal to 70 .5.

01:33 And this is going to be approximately equal using the approximation to the probability that z is less than or equal to 70 .5 minus n times p over the square root of n times p times 1 minus p.

02:06 This is the probability that z is less than or equal to 70 .5 minus 80, which we've already solved for here, divided by 80 times 0 .6.

02:27 This is all square root of.

02:34 It's equal to the probability that z is less than or equal to negative 1 .3712, and this is equal to approximately 0 .08 5 -2.

02:56 For part b we want the probability that x is greater than 70 and less than 90.

03:14 Now this is equal to the probability that x is less than 90 minus the probability that x is less than or equal to 70.

03:30 Now i'm making sure to get this less than or equal correct because it matters when we apply the continuity correction factor...

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