Respuesta :
Answer:
13.57% probability that more than 10 new users will sign up for the Lookbook social networking site in the next minute
Step-by-step explanation:
To solve this question, we need to understand the poisson and the normal probability 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:
[tex]P(X = x) = \frac{e^{-\lambda}*\lambda^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\lambda[/tex] is the mean in the given interval. The variance is the same as the mean.
Normal probability distribution:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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.
To approximate the Poisson distribution, we use [tex]\mu = \lambda, \sigma = \sqrt{\lambda}[/tex]
On average, 7.5 users sign up for Lookbook each minute
This means that [tex]\lambda = 7.5[/tex]. So
[tex]\mu = 7.5, \sigma = \sqrt{7.5} = 2.7386[/tex]
What is the probability that more than 10 new users will sign up for the Lookbook social networking site in the next minute
Using continuity correction, this is P(X > 10 + 0.5) = P(X > 10.5), which is 1 subtracted by the pvalue of Z when X = 10.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{10.5 - 7.5}{2.7386}[/tex]
[tex]Z = 1.1[/tex]
[tex]Z = 1.1[/tex] has a pvalue of 0.8643.
1 - 0.8643 = 0.1357
13.57% probability that more than 10 new users will sign up for the Lookbook social networking site in the next minute
