About 4% of the population has a particular genetic mutation. 700 people are randomly selected.
Find the standard deviation for the number of people with the genetic mutation in such groups of 700.
To find the standard deviation for the number of people with the genetic mutation in a group of 700, we need to first calculate the variance and then take the square root of the variance.
Step 1: Calculate the expected value
The expected value is the average number of people with the genetic mutation in a group of 700. Since 4% of the population has the mutation, the expected value can be calculated as follows:
Expected Value = (4%)(700) = 0.04 * 700 = 28
Step 2: Calculate the variance
To calculate the variance, you need to find the difference between each observed value (0 to 700) and the expected value (28), square each difference, and then sum up all the squared differences. The formula for variance is given by:
Variance = ∑(x - E(X))^2 * P(x)
Where:
(x - E(X)) is the difference between the observed value and the expected value
P(x) is the probability of that observed value
In this case, P(x) for each value is 0.04 (since the probability of having the genetic mutation is still 4% regardless of the observed value). So the variance can be calculated as follows:
Variance = ∑(x - 28)^2 * 0.04
To calculate this by hand, you would need to substitute each observed value (0 to 700) into the formula, square the difference (x - 28)^2, and multiply it by 0.04. Sum up all these values to get the variance. However, as this involves numerous calculations, it is more practical to use statistical software (e.g., Microsoft Excel, R, or Python) to do the calculations for you.
Step 3: Calculate the standard deviation
After finding the variance, take the square root of the variance to get the standard deviation. The standard deviation is a measure of how spread out the data is from the expected value.
Standard Deviation = √Variance
Again, you can use a statistical software or a calculator with a square root function to find the standard deviation.
Note: In this scenario, assuming that each observation is independent, we can apply the binomial distribution with parameters n = 700 and p = 0.04 to estimate the number of people with the genetic mutation. The variance and standard deviation formulas mentioned above can be used for a binomial distribution as well.