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Chapter 24: Problem 872

The packaging of breakfast cereals is done automatically. Each package is filled with a certain number of grams of cereal. Assume a package is filled with \(400 \mathrm{~g}\) of cereal and is labeled as such. The cereal processor has kept a record of his variation and found the population standard deviation to be \(\sigma=14 \mathrm{~g} .\) He knows that he will be satisfying the "truth-in-packaging" law if the variation about the printed weight on the package is held within 2 standard deviations \(95 \%\) of the time. How can he determine whether \(95 \%\) of his packages will fall within \(400 \mathrm{~g} \pm 2 \sigma ?\)

Short Answer

Expert verified
The cereal processor can determine if 95% of his packages meet the "truth-in-packaging" law by using the empirical rule for normal distributions, which states that approximately 95% of the data should fall within 2 standard deviations of the mean. In this case, the acceptable weight limits are \(372g\) to \(428g\) (which corresponds to \(400g \pm 2(14g)\)). Since these limits represent 95% of the packages' weights, the cereal processor will satisfy the law.

Step by step solution

01

Write down the problem data

First, let's write down the given information: - Mean weight: \(\mu = 400g\) - Population standard deviation: \(\sigma = 14g\) - Percentage required to fall within acceptable limits: \(95\%\) - The acceptable weight limits are 400g 卤 2 standard deviations (\(2\sigma\))
02

Determine the acceptable weight limits

To find the acceptable weight limits, we need to add and subtract 2 times the standard deviation from the mean weight: Lower limit: \(400 - 2(14) = 400 - 28 = 372g\) Upper limit: \(400 + 2(14) = 400 + 28 = 428g\) So, the acceptable weight limits are between 372g and 428g.
03

Apply the empirical rule

Using the empirical rule (also called the 68-95-99.7 rule) for normal distributions, approximately: - 68% of the data falls within 1 standard deviation (\(\sigma\)) of the mean (\(\mu\)) - 95% of the data falls within 2 standard deviations (\(2\sigma\)) of the mean (\(\mu\)) - 99.7% of the data falls within 3 standard deviations (\(3\sigma\)) of the mean (\(\mu\)) In this exercise, we need to check if 95% of the packages fall within the acceptable weight limits (400g 卤 2 standard deviations).
04

Determine if the conditions are met

Since we are interested in the 95% range, the empirical rule tells us that about 95% of the data should fall within 2 standard deviations of the mean weight. In this case, the acceptable weight limits are 372g to 428g, which corresponds to 2 standard deviations from the mean (400g 卤 2(14g)). As such, the cereal processor will be satisfying the "truth-in-packaging" law because 95% of the packages are expected to fall within 400g 卤 2 standard deviations, which represents 95% of the packages' weights.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution
A normal distribution, often referred to as a bell curve, is a fundamental concept in statistics and probability theory. It is a continuous probability distribution characterized by its symmetrical, bell-shaped curve. A normal distribution is defined by two parameters: the mean (\(\mu\)) and the standard deviation (\(\sigma\)).
  • The mean is the central point of the distribution, and it represents the average of all data points.
  • The standard deviation indicates the spread or dispersion of the data around the mean.
Normal distributions are important because many natural phenomena follow this pattern, and they provide a basis for inferential statistics. In the exercise mentioned, we assume that the weight of cereal packages is normally distributed around a mean with a certain standard deviation.
Standard Deviation
Standard deviation (\(\sigma\)) is a statistic that measures the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a broader range. It is essential in assessing the reliability and consistency of data.The formula for standard deviation is:\[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}\]where \(N\) is the number of observations, \(x_i\) represents each observation, and \(\mu\) is the mean of the data. In the context of the exercise, the standard deviation of 14g tells us how much the weight of the cereal packages varies from the mean weight. This parameter is vital for determining the acceptable weight limits around the mean using the empirical rule.
Truth-in-Packaging Law
The truth-in-packaging law ensures that consumers receive the amount of product as described on the packaging. This law is vital for maintaining fairness and transparency in consumer transactions. The law dictates that a certain percentage of packages must contain at least the advertised weight. In this exercise, the cereal processor wants to comply with the truth-in-packaging law by ensuring that 95% of the packages fall within 2 standard deviations of the mean weight. If 95% of packages meet this requirement, it indicates compliance with the law, thus ensuring that most consumers receive the correct amount of cereal.
Mean Weight
The mean weight, represented by \(\mu\), is a measure of the central tendency or 'average' of a set of data points. It is calculated by summing up all the values and dividing by the number of values.For example, if you have a set of cereal packages with weights, the mean weight would be the total weight of all packages divided by the number of packages.In statistical terms:\[\mu = \frac{1}{N}\sum_{i=1}^{N} x_i\]where \(N\) is the number of data points, and \(x_i\) represents each individual data point.In this exercise, the mean weight is 400g, which serves as the baseline around which the standard deviation is applied to determine the acceptable range of weights for the packaging.

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Most popular questions from this chapter

Human mortality increases almost exponentially once a person reaches 25 years of age. Using this empirical fact, derive the distribution function and probability density function associated with the Gompertz law found in actuarial science.

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Twenty cars from two comparable major brand models are selected at random. These 40 cars are subjected to accelerated life testing (they are driven many kilometers over very poor roads in a short time) and their failure times are recorded in weeks: $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|} \hline \text { Brand U: } & 25 & 31 & 20 & 42 & 39 & 19 & 35 & 36 & 44 & 26 \\\ & 38 & 31 & 29 & 41 & 43 & 36 & 28 & 31 & 25 & 38 \\ \hline \text { Brand V: } & 28 & 17 & 33 & 25 & 31 & 21 & 16 & 19 & 31 & 27 \\\ & 23 & 19 & 25 & 22 & 29 & 32 & 24 & 20 & 34 & 26 \\ \hline \end{array} $$ Determine whether the two brands of cars have equal distributions for the length of their lives under accelerated life testing.
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