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Answer:
P(X ≥ 125) = 0.9812
Step-by-step explanation:
To solve this question, we need to understand the Poisson distribution and the normal distribution.
Poisson distribution:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
In which
x is the number of sucesses
e = 2.71828 is the Euler number
is the mean in the given interval, which is the same as the variance.
Normal distribution:
When the distribution is normal, we use the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
The Poisson can be approximated to the normal, with
Let μ = 2.5 every minute
This is the mean of the Poisson, so , in which n is the number of minutes.
P(X ≥ 125) over an hour
An hour has 60 minutes, so
Using continuity correction, this is , which is 1 subtracted by the pvalue of Z when X = 124.5. So
has a pvalue of 0.0188
1 - 0.0188 = 0.9812
So
P(X ≥ 125) = 0.9812