The packaging process in a breakfast cereal company has been adjusted so that an average of 13.0 oz of cereal is placed in each

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The packaging process in a breakfast cereal company has been adjusted so that an average of 13.0 oz of cereal is placed in each

package. Of course, not all packages have precisely 13.0 oz because of random sources of variability. The standard deviation of the actual net weight is 0.1 oz, and the distribution of weights is known to follow the normal probability distribution. Determine the probability that a randomly chosen package will contain between 13.0 and 13.2oz of cereal and illustrate the proportion of area under the normal curve which is associated with this probability value.

Mathematics

1 answer:

PIT_PIT [12.4K] · Answer 1 · 2026-03-14T16:37:49+03:00

In order to determine this probability, we calculate using this difference: To obtain these probabilities, it’s possible to utilize normal standard distribution tables, a calculator, or software like Excel. The accompanying figure displays the results achieved. Here’s a detailed breakdown of the steps: Relevant concepts include the normal distribution, which describes a probability distribution that is symmetric regarding the mean, demonstrating that occurrences close to the mean are more likely than those farther away. The Z-score represents a statistical measure illustrating how far a value is from the average of a set, expressed in standard deviations. For our analysis, let X denote the random variable representing weights in a population, with its distribution characterized by: We’re specifically interested in this probability. The most effective approach to address this issue is through the standard normal distribution and the Z-score calculation, expressed as: Applying this formula to our probability provides the following: This allows us to calculate this probability with the provided difference: We use standard distribution tables, a calculator, or Excel for determining these probabilities. The graph illustrates the resulting outcome.