Given a random sample size of n=900 from a binomial...

Given a random sample size of $n=900$ from a binomial probability distribution with $P=0.10$ do the following: a. Find the probability that the number of successes is greater than $110 .$ b. Find the probability that the number of successes is fewer than 53 . c. Find the probability that the number of successes is between 55 and 120 . d. With probability $0.10$, the number of successes is fewer than how many? e. With probability $0.08$, the number of successes is greater than how many?

Given a random sample size of $n=900$ from a binomial probability distribution with $P=0.10$ do the following:
a. Find the probability that the number of successes is greater than $110 .$
b. Find the probability that the number of successes is fewer than 53 .
c. Find the probability that the number of successes is between 55 and 120 .
d. With probability $0.10$, the number of successes is fewer than how many?
e. With probability $0.08$, the number of successes is greater than how many?

Key Concepts

Binomial Distribution

The binomial distribution represents the probability of obtaining a fixed number of successes in a fixed number of independent trials, each with the same probability of success. In this context, the problem involves a binomial distribution with parameters n (number of trials) and p (success probability), which defines both the mean (np) and variance (np(1-p)).

Normal Approximation

When the number of trials n is large, the binomial distribution can be approximated by a normal distribution. This approximation is valid provided that both np and n(1-p) are sufficiently large, allowing the use of the normal distribution's properties to estimate binomial probabilities.

Continuity Correction

The continuity correction is used when approximating a discrete distribution (like the binomial) with a continuous one (like the normal). It involves adjusting the binomial counts by 0.5 to more accurately estimate probabilities of exact outcomes and intervals.

Z-score Transformation

The Z-score transformation standardizes values from a normal distribution by subtracting the mean and dividing by the standard deviation. This transformation allows one to utilize standard normal distribution tables or software to find probabilities associated with various outcomes.

Inverse Probability and Quantiles

Finding quantiles (or using inverse probability) involves determining the value below (or above) which a certain percentage of the observations in a probability distribution fall. In this problem, quantiles are used to answer questions about the number of successes corresponding to given tail probabilities.

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