A company manufactures calculators in batches of 64 and there is a 5% rate of defects. What is the standard deviation for the number of defects per batch?

Standard deviation:

sd = √npq = √(64)(.05)(.95) = ?

Note: q = 1 - p

I'll let you finish the calculation.

0.95

To calculate the standard deviation for the number of defects per batch, we need to use the formula for the standard deviation for a binomial distribution:

σ = √(n * p * q)

Where:
- σ is the standard deviation
- n is the number of trials (number of batches)
- p is the probability of success (defects)
- q is the probability of failure (no defects)

In this case, the number of trials (batches) is 64, the probability of defects (success) is 5% or 0.05, and the probability of no defects (failure) is 1 - 0.05 = 0.95.

Plugging the values into the formula, we have:

σ = √(64 * 0.05 * 0.95)

Calculating this using a calculator:

σ = √(3.84)

σ ≈ 1.96

So, the standard deviation for the number of defects per batch is approximately 1.96.

To calculate the standard deviation for the number of defects per batch, we need to know the mean and the variance.

First, let's calculate the mean. The mean number of defects per batch can be found by multiplying the total number of calculators in a batch (64) by the defect rate (5% or 0.05). So the mean (µ) is:

µ = 64 * 0.05 = 3.2

Next, we need to calculate the variance. The variance is the squared deviation from the mean, multiplied by the probability of each possible outcome. In this case, the possible outcomes are zero defects (with a probability of 1 - the defect rate) and one defect (with a probability equal to the defect rate).

So the variance (σ^2) is:

σ^2 = [(0 - 3.2)^2 * (1 - 0.05)] + [(1 - 3.2)^2 * 0.05]

Calculating this out, we get:

σ^2 = (10.24 * 0.95) + (5.76 * 0.05) = 9.728 + 0.288 = 10.016

Finally, we can find the standard deviation by taking the square root of the variance:

σ = √(10.016) ≈ 3.165

Therefore, the standard deviation for the number of defects per batch is approximately 3.165.