Question

Can some one help me on this questions?

In checking river water samples for bacteria, water is placed in a culture medium so that certain bacteria colonies can grow if those bacteria are present. The number of colonies per dish averages 12 for water samples from a certain river.
a. Find the probability that the next dish observed will have at least 10 colonies.
b. Find the mean and standard deviation of the number of colonies per dish.

Answer 1

Are we to assume some applicable distribution?

Answer 2

a)I would suggest to use Poisson distribution with lambda=12 colonies/dish
\[P(n \ge 10)=1-P(n <10)=1-\sum_{k=0}^{9}\frac{ \lambda^k }{ k! }e^{-\lambda}\]The result I get is P(n>=10)=1-0.24=0.76
b)For Poisson distribution: mean=lambda = 12 and variance =lambda, which means the standard deviation amounts to square root of 12

To solve a) you may also use the normal approximation where the z-score can be calculated as:\[Z=\frac{ X-\lambda }{ \sqrt{\lambda} }\]\[P(X>9.5)=1-P(X \le 9.5)=1-P(Z \le\frac{ 9.5-12 }{ \sqrt{12} })=1-0.24=0.76\]

Answer 3

yeah thats what i was thinking ^^

Answer 4

I wouldn't use the Normal at all. Just not enough information about the variance.

Answer 5

i disagree, assuming its a poisson distribution with mean 12, then Normal dist is a good approximation since lambda > 10

http://en.wikipedia.org/wiki/Poisson_distribution#Related_distributions

http://www.wolframalpha.com/input/?i=sum+12%5Ek+*e%5E%28-12%29%2Fk%21+for+k+%3D0+to+9

Answer 6

THANK'S FOR ALL

Answer 7

@dumbcow Generally I agree, but the Poisson was a big enough stretch. To approximate THAT and make it a bigger stretch just makes me a little nervous.

Thanks for the discussion.

Answer 8

Agree with @dumbcow. Precisely I have solved for Poisson and for Normal approximation leading to same results