Answered: In this problem, we will approximate a…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Counting And Probability

Section9.3: Binomial Probability

Problem 2E: If a binomial experiment has probability p success, then the probability of failure is...

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Please help with problems a, b, and c:

In this problem, we will approximate a binomial dis- tribution by a normal distribution. From looking at the pmfs of the binomial distribution, for large values of n the histogram of the binomial pmf looks “bell-shaped”. It turns out that the normal distribution is a good approximation for the binomial approximation for large n. It also works for small n as long as p is not too far from 0.5. For convenience, we will let q = 1 − p.

(a) Let X~ Binomial (25, 0.4). Compute P(X = 10).
(b) We will approximate with a normal random variable. Let Y
be a normal random variable with the same mean and variance
as X. Compute P(9.5 ≤ Y ≤ 10.5). We will use this as an
approximation for P(X = 10).
(c) Briefly explain why we used the event (9.5 ≤ Y ≤ 10.5) to
approximate the event (X = 10).

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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