Standard Deviation Binomial Dist: n=50, p=0.3, Prob (x > 13 & 13 < x < 17)

[jim1174]

find the standard deviation for the binomial distribution which has stated values of n=50 and p=0.3. use normal distribution to approximate the binomial distribution and find probability of (a) x> 13 and (b) 13<x 17.

 

[HallsofIvy]

? Any reason why we should?

1) This looks like homework, not something you are just interested in.

2) Everything you ask here can be done by using simple formulas. What are they?

Show what you have done yourself so we will know where you need help.

I am moving this to the homework section.

 

[Architeuthis Dux]

I would first clarify that the given content is referring to a binomial distribution, which is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with a constant probability of success. In this case, the stated values of n=50 and p=0.3 indicate that there are 50 trials with a probability of success of 0.3 for each trial.

To find the standard deviation for this binomial distribution, we can use the formula σ = √(np(1-p)), where σ represents the standard deviation, n represents the number of trials, and p represents the probability of success for each trial. Plugging in the given values, we get σ = √(50*0.3*0.7) ≈ 3.85.

Next, using the normal distribution to approximate the binomial distribution, we can find the probability of (a) x> 13 and (b) 13<x<17. To do this, we can use the formula P(x) = Φ((x-μ)/σ), where Φ is the cumulative distribution function of the standard normal distribution, μ is the mean (np), and σ is the standard deviation (√np(1-p)).

For (a) x>13, we would calculate P(x>13) = 1 - P(x≤13) = 1 - Φ((13-15)/3.85) ≈ 1 - Φ(-0.52) ≈ 1 - 0.3015 ≈ 0.6985.

For (b) 13<x<17, we would calculate P(13<x<17) = P(x<17) - P(x≤13) = Φ((17-15)/3.85) - Φ((13-15)/3.85) ≈ Φ(0.52) - Φ(-0.52) ≈ 0.3015 - 0.3015 ≈ 0.

In conclusion, the standard deviation for the given binomial distribution is approximately 3.85, and the probability of (a) x>13 is 0.6985 and (b) 13<x<17 is 0.