Question

The weights of a certain brand of candies are normally distributed with a mean weight of 0.8582 g and a standard deviation of 0.0516 g. A sample of these candies came from a package containing 457 candies, and the package label stated that the net weight is 390.3 g. (If every package has 457 candies, the mean weight of the candies must exceed

457/390.3=0.8541 g for the net contents to weigh at least 390.3 g )
a. If 1 candy is randomly selected, find the probability that it weighs more than 0.8541 g. The probability is (Round to four decimal places as needed.)
b. If 457 candies are randomly selected, find the probability that their mean weight is at least 0.85419 . The probability that a sample of 457 candies will have a mean of 0.8541g or greater is (Round to four decimal places as needed.)
c. Given these results, does it seem that the candy company is providing consumers with the amount claimed on the label? because the probability of getting a sample mean of 0.8541 g or greater when 457 candies are selected exceptionally small

Answer 1

The question involves calculating probabilities using the normal distribution, assessing sample standard deviations to determine the need for equipment recalibration, and evaluating whether a candy company's package labeling is accurate based on statistical evidence.

Step-by-step explanation:

The problem deals with the normal distribution of candy weights and uses statistical methods to determine the likelihood that a sample of candies will meet a specified weight criterion. Probability is calculated using the Z-score formula for individual candy weight and for the mean weight of a sample of candies. Additionally, the problem asks to consider if the observed sample standard deviation from a machine indicates a need for recalibration, and whether a company's packaging is accurately labeled based on the probability of sample sums.

• For individual candies, we calculate the probability that one candy weighs more than a set amount using the Z-score and the normal distribution probability function.
• For a sample of candies, the central limit theorem is used to find the probability that their mean weight is at least a particular value.
• The need for machine recalibration is assessed based on the calculated standard deviation of a sample compared to expected variability.

Considering these statistical concepts, if the probability of a sample of candies having a certain mean weight is very high, we can be confident that the candy company is legitimately providing the amount claimed on their labels.