Probability of Sample Mean for Poisson Distribution

• Unusualskill
In summary, the problem states that a field with a grid of 1m squares has a Poisson distribution with mean 2.25 snails per square meter. A random sample of 120 squares is observed and the probability of the sample mean being at most 2.5 snails is to be found using the central limit theorem. The solution uses the formula for a normal distribution with a mean of 2.25 and a standard deviation of 0.01875. The probability is calculated to be approximately 0.9682. The solution is correct, but it is unclear whether or not the Poisson distribution can be used in this way for sampling.
Unusualskill

Homework Statement

A rectangular field is gridded into squares of side 1m. at one time of the year the number of snails in the field can be modeled by a Poisson distribution with mean 2.25 per m^2.
(i) a random sample of 120 squares is observed and the number of snails in each square counted. find the probability that the sample mean number of snails is at most 2.5

Homework Equations

using central limit theorem, sample mean of X ~N (2.25, 2.25x1/120=0.01875)

p(sample mean of x<=2.5)
=p(z<(2.5+1/240-2.25)/ √0.01875)
=p(z<1.856)
=09682

The Attempt at a Solution

Is this correct? any1? becuz I am nt sure whether poisson can do this way onot in sampling

Unusualskill said:

Homework Statement

A rectangular field is gridded into squares of side 1m. at one time of the year the number of snails in the field can be modeled by a Poisson distribution with mean 2.25 per m^2.
(i) a random sample of 120 squares is observed and the number of snails in each square counted. find the probability that the sample mean number of snails is at most 2.5

Homework Equations

using central limit theorem, sample mean of X ~N (2.25, 2.25x1/120=0.01875)

p(sample mean of x<=2.5)
=p(z<(2.5+1/240-2.25)/ √0.01875)
=p(z<1.856)
=09682

The Attempt at a Solution

Is this correct? any1? becuz I am nt sure whether poisson can do this way onot in sampling

Please write proper english here; any1 = anyone, I am = I'm, etc, etc. After you fix that up I will be glad to tell you whether or not I agree with your solution.

1. What is the Poisson distribution?

The Poisson distribution is a probability distribution that is used to model the number of events that occur in a fixed interval of time or space.

2. How is the Poisson distribution different from other probability distributions?

The Poisson distribution is unique in that it models the number of events that occur in a fixed interval, rather than the probability of a single event occurring.

3. What is the formula for the Poisson distribution?

The formula for the Poisson distribution is P(X = k) = (λ^k * e^-λ)/ k!, where λ is the average number of events in the interval and k is the number of events.

4. What types of data can be modeled using the Poisson distribution?

The Poisson distribution is commonly used to model data that involves counts, such as the number of customers in a store, number of accidents on a highway, or number of emails received per day.

5. How is the Poisson distribution used in real-world applications?

The Poisson distribution is used in a variety of fields, including finance, engineering, and epidemiology, to analyze and predict the occurrence of events. For example, it can be used to estimate the number of insurance claims that will be filed in a given time period or the number of defects in a manufacturing process.

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