The number of nuts in a package of Nutty's premium cashews is normally distributed with mean of 432.4 nuts and standard deviation of 5.8 nuts packages with fewer than 420 nuts or more than 445 nuts will be rejected by quality control
a) what is the probability that a package selected at random will fall within the acceptable quality control range?
b) in a production run of 50,000 packages of nuts, how many can be expected to be rejected?
a. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to those Z scores.
b. Sum the two proportions and multiply by 50,000.
a) To find the probability that a package selected at random will fall within the acceptable quality control range, we need to find the area under the normal distribution curve between 420 and 445.
Step 1: Calculate the z-scores for the lower and upper bounds.
The z-score is a measure of how many standard deviations an observation is from the mean. It is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
For the lower bound (420), the z-score is:
z1 = (420 - 432.4) / 5.8
For the upper bound (445), the z-score is:
z2 = (445 - 432.4) / 5.8
Step 2: Look up the corresponding probabilities.
Using a standard normal distribution table or a calculator, find the probabilities associated with the z-scores calculated in step 1. The probabilities are the areas under the curve between the z-scores.
Step 3: Calculate the probability between the lower and upper bounds.
Subtract the probability obtained for the lower bound from the probability obtained for the upper bound. This gives us the probability that a random package falls within the acceptable quality control range.
b) To determine how many packages can be expected to be rejected in a production run of 50,000 packages, we need to calculate the expected number of packages that fall outside the acceptable quality control range.
Step 1: Calculate the probability of a single package being rejected.
The probability of a single package being rejected is equal to 1 minus the probability of falling within the acceptable quality control range (found in part a).
Step 2: Multiply the probability of rejection by the total number of packages.
Multiply the probability of rejection by the total number of packages (50,000) to find the expected number of packages that will be rejected.