Use the normal distribution to approximate the following Binomial probability. A product is manufactured in batches of 100 and a defect rate of 8%. Find the probability of more then 7 defects.
To approximate the probability of more than 7 defects using the normal distribution, we need to calculate the mean (μ) and standard deviation (σ) of the binomial distribution first.
The mean (μ) of a binomial distribution is given by: μ = np, where n is the number of trials and p is the probability of success.
In this case, the number of trials is 100 (batches of 100) and the probability of success (defect) is 8% or 0.08. Therefore, μ = 100 * 0.08 = 8.
The standard deviation (σ) of a binomial distribution is given by: σ = √(np(1-p)).
Plugging in the values, σ = √(100 * 0.08 * (1 - 0.08)) = √(6.4 * 0.92) ≈ √5.888 ≈ 2.425.
Now that we have the mean (μ = 8) and standard deviation (σ ≈ 2.425), we can use the properties of the normal distribution to approximate the probability of more than 7 defects.
To do this, we need to calculate the z-score corresponding to 7 defects. The z-score (z) is given by: z = (x - μ) / σ, where x is the number of defects.
For "more than 7 defects" we need to calculate the probability of x > 7, which can be approximated by finding the area under the normal curve to the right of the z-score corresponding to 7 defects.
Using the z-score formula, the z-score for 7 defects is: z = (7 - 8) / 2.425 ≈ -0.412.
To find the probability that x > 7, we can use a standard normal distribution table or a statistical calculator to find the area to the right of the z-score (-0.412).
Alternatively, we can use a calculator, such as the "normcdf" function in Excel or the "pnorm" function in R, to directly calculate the probability.
For example, using the "normcdf" function in Excel, we can type "=1-normcdf(7, 8, 2.425)" to calculate the probability that x > 7.
Therefore, the probability of having more than 7 defects in a batch of 100 is approximately the calculated value from the standard normal distribution table or the result obtained using the "normcdf" function in Excel.