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Chapter 7: Problem 28

Explain why the weight of a package of one dozen tomatoes should be approximately normally distributed if the dozen tomatoes represent a random sample.

Short Answer

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Answer: The weight of a package of one dozen tomatoes should be approximately normally distributed because, according to the Central Limit Theorem, the distribution of sample means (the average weight of each tomato in the package) will approach a normal distribution when taking several random samples. Since the sum of the individual weights of the twelve tomatoes also follows a normal distribution, the total weight of a package with one dozen tomatoes should be approximately normally distributed.

Step by step solution

01

Understanding Normal Distribution

A normal distribution, also known as a Gaussian distribution or bell curve, is a probability distribution that is symmetric around the mean, showing that data near the mean is more frequent in occurrence than data far from the mean. In essence, this means that most of the observations in a dataset are expected to be close to the average value and fewer observations are expected to be farther away from the average value.
02

Central Limit Theorem

The Central Limit Theorem (CLT) states that when we take a large enough random sample from a population, regardless of the original distribution, the distribution of the sample means will approach a normal distribution. This means that the more samples are taken, the more the distribution of the samples' average values will resemble a normal distribution.
03

Applying Central Limit Theorem to Package of Tomatoes

In this case, we have a random sample of one dozen tomatoes. If we take several samples of one dozen tomatoes, each sample's average weight will be closer to the normal distribution according to the Central Limit Theorem.
04

Package Weight as an Approximation of Normal Distribution

The weight of a package containing one dozen tomatoes can be considered as the sum of the individual weights of those twelve tomatoes. Since these tomatoes are randomly sampled from a larger population of tomatoes, and due to the Central Limit Theorem, the distribution of the sample means (the average weight of each tomato in the package) will approach a normal distribution. The sum of weights with a normal distribution will also follow a normal distribution, so the total weight of the package with one dozen tomatoes should be approximately normally distributed.

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Most popular questions from this chapter

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