Problem 19 - Algebra 2

Question

Agriculture To win a prize, a tomato must be greater than 4 in. in diameter. The diameters of a crop of tomatoes grown in a special soil are normally distributed, with a mean of 3.2 in. and a standard deviation of 0.4 . Find the probability that the crop will contain a winning tomato.

Step-by-Step Solution

Answer

The probability that the crop will contain a winning tomato is calculated by subtracting the Z-table probability for 4 inches from 1. This results in the probability that a tomato will have a diameter greater than 4 inches.

Step 1: Understanding the Problem

First identify the details given in the problem. We are dealing with a normal distribution where the mean diameter (\( \mu \)) is 3.2 inches and the standard deviation (\( \sigma \)) is 0.4 inches. We need to find the probability of a tomato having a diameter greater than 4 inches.

Step 2: Calculating the z-score

We need to calculate the z-score for a tomato with a diameter of 4 inches. The z-score is calculated using the formula \( z = \frac{x-\mu}{\sigma} \) where \( x \) is the value for which we are finding the z-score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Step 3: Look Up the Probability in Z-table

After calculating the z-score, the next step is to find the corresponding probability from a standard normal distribution table (also known as a z-table). This table gives the probability that a normally distributed random variable is less than a given value.

Step 4: Find the Complementary Probability

Since we are finding the probability of the diameter being greater than a certain size, we need to subtract the probability found from 1 to get the complementary probability.

Key Concepts

Understanding the Z-Score

The z-score is a fundamental concept in statistics, especially when dealing with a normal distribution. Essentially, the z-score is a way to standardize values, allowing us to see where a particular value falls within the distribution. It represents how many standard deviations away a specific value (like our tomato's diameter of 4 inches) is from the mean. The formula for calculating the z-score is:

• \( z = \frac{x - \mu}{\sigma} \)

Here:

• \( x \) is the value we're analyzing, which is 4 inches in this case.
• \( \mu \) is the mean of the distribution, given as 3.2 inches.
• \( \sigma \) is the standard deviation, which is 0.4 inches.

By calculating the z-score, we transform our data into the standard normal distribution, which has a mean of 0 and a standard deviation of 1. This makes it much easier to work with because we have a universal scale to determine probabilities using z-tables or statistical software.

Decoding Standard Deviation

Standard deviation tells us how much variation or spread there is in a set of data points. In our tomato problem, the standard deviation is 0.4 inches. This number provides an idea of how much the tomato diameters generally deviate from the mean diameter of 3.2 inches. A small standard deviation means that the values tend to be close to the mean, while a larger one indicates that the data points are spread out over a wider range. Standard deviation is used to calculate the z-score, which in turn helps us figure out probabilities in a normal distribution. By understanding how spread out our diameters are, we can better understand the probability of a tomato reaching the winning size. In practical terms:

• A lower standard deviation means consistent sizing, with data points clustered around the mean.
• A higher standard deviation means more variability, indicating that diameters could vary significantly from 3.2 inches.

Knowing the standard deviation gives us insight into the variability of our data.

The Role of the Mean in Normal Distribution

The mean is a central concept in statistics and refers to the average of all the values in a data set. In the normal distribution of our tomato diameters, the mean is 3.2 inches. This figure gives us the expected average size of the tomatoes from the crop grown in special soil. The mean is the peak of the bell curve in a normal distribution, showing where most of the data points tend to cluster. Understanding the mean is crucial because it serves as the balancing point of the distribution, providing a point from which all deviations (measured by the standard deviation) are calculated. Knowing the mean allows us to assess whether a particular tomato is substantially larger or smaller than what is typical for our data set.

• It's the point of reference for calculating z-scores.
• It helps us interpret the standard deviation, indicating the typical range of tomato sizes.

By using the mean, we can determine just how rare a winning tomato might be and calculate the probability of encountering such an exceptional tomato size in the crop.