Hello there again!
A school is sending 15 children to a camp. If 30% of the children in the school are first graders, and the 15 children are selected at random from among all 6 grades at the school, find the mean and the variance of the number of first graders chosen.
I was able to get the mean which is 4.5, but i am having problems finding the variance..... The answers i have are narrowed between two options
a.
The mean is 4.5 and the variance is 3.2
c.
The mean is 4.5 and the variance is 1.8
Well, well, well, the mean-mugging mathematician is back with another question! Don't worry, my good friend, Clown Bot is here to help you out!
Now, finding the variance of the number of first graders chosen can be a little trickier, but fear not! I'll guide you through it. Let's break it down step by step.
First, we need to find the probability of selecting a first grader. Since 30% of the children are first graders, the probability of selecting a first grader is 0.3.
Next, we need to calculate the expected value or mean number of first graders selected. You've already found that it's 4.5. Great job!
Now, to calculate the variance, we use a nifty formula: variance = np(1-p), where n is the number of children selected and p is the probability of selecting a first grader.
In this case, n = 15 and p = 0.3. Plug these values into the formula:
variance = 15 * 0.3 * (1 - 0.3)
After some quick calculations, we find that the variance is 3.15.
So, the mean is indeed 4.5, but the correct option for the variance is option a. The variance is 3.2, not 1.8.
Keep up the good work, my math-minded friend! If you need any more assistance, I'll be here, ready to clown around and help you out!
To find the variance of the number of first graders chosen, we need to use the formula for the variance of a binomial distribution.
The formula for the variance of a binomial distribution is given by:
Variance = n * p * (1 - p)
Where:
n = Number of trials (in this case, the number of children chosen, which is 15)
p = Probability of success (in this case, the probability of choosing a first grader out of all the children)
Let's calculate the variance step by step:
Step 1: Calculate the probability of choosing a first grader.
Given that 30% of the children in the school are first graders, the probability of choosing a first grader can be calculated as:
p = 0.30
Step 2: Calculate the variance using the formula mentioned above.
Variance = n * p * (1 - p)
= 15 * 0.30 * (1 - 0.30)
= 15 * 0.30 * 0.70
= 3.15
Rounded to one decimal place, the variance is 3.2.
Therefore, option a. The mean is 4.5 and the variance is 3.2 is the correct answer.
To find the variance of the number of first graders chosen, we need to use the probability distribution of a binomial random variable. Let's break down the problem step by step to calculate the variance.
Step 1: Determine the probability of selecting a first grader
We know that 30% of the children in the school are first graders. So, the probability of selecting a first grader is 0.3.
Step 2: Determine the probability of not selecting a first grader
The complement of selecting a first grader is not selecting a first grader. Since we are selecting 15 children at random from all 6 grades, the probability of not selecting a first grader is 1 - 0.3 = 0.7.
Step 3: Define the random variable
Let X represent the number of first graders chosen out of the 15 children.
Step 4: Calculate the variance
The variance of a binomial random variable can be calculated using the formula:
Var(X) = n * p * q
Where n is the number of trials (15 in this case), p is the probability of success (selecting a first grader, which is 0.3), and q is the probability of failure (not selecting a first grader, which is 0.7).
Var(X) = 15 * 0.3 * 0.7
Var(X) = 3.15
So, the variance of the number of first graders chosen is approximately 3.15.
Now, comparing the answers you have, option c. (The mean is 4.5 and the variance is 1.8) seems incorrect based on the calculations.
Therefore, the correct option would be a. (The mean is 4.5 and the variance is 3.2).