Problem 36 - Calculus for Biology and Medicine

Question

The mean weight of goats on a farm is \(123 \mathrm{lb}\), and the standard deviation is \(9 \mathrm{lb}\). If the weights are normally distributed, determine what percentage of goats weigh (a) between 110 and \(130 \mathrm{lb}\), (b) less than \(100 \mathrm{lb}\), and \((\mathbf{c})\) more than \(150 \mathrm{lb}\).

Step-by-Step Solution

Answer

(a) 70.74%, (b) 0.52%, (c) 0.13%

Step 1: Identify Given Information

We are given the mean weight of goats as \( \mu = 123 \mathrm{lb} \) and the standard deviation as \( \sigma = 9 \mathrm{lb} \). The weights follow a normal distribution.

Step 2: Calculate Z-scores

For each part, calculate the z-scores using the formula:\[ z = \frac{x - \mu}{\sigma} \]- For part (a), the z-scores for 110 lb and 130 lb are: - \( z_{110} = \frac{110 - 123}{9} = -1.44 \) - \( z_{130} = \frac{130 - 123}{9} = 0.78 \)- For part (b), the z-score for 100 lb is: - \( z_{100} = \frac{100 - 123}{9} = -2.56 \)- For part (c), the z-score for 150 lb is: - \( z_{150} = \frac{150 - 123}{9} = 3.00 \)

Step 3: Use the Standard Normal Table

Use the standard normal distribution table (or a calculator) to find the cumulative probabilities:- Part (a): - \( P(z < 0.78) \approx 0.7823 \) - \( P(z < -1.44) \approx 0.0749 \) - Probability between 110 and 130 lb is \( 0.7823 - 0.0749 = 0.7074 \)- Part (b): - \( P(z < -2.56) \approx 0.0052 \)- Part (c): - \( P(z < 3.00) \approx 0.9987 \) - Therefore, \( P(z > 3.00) = 1 - 0.9987 = 0.0013 \)

Step 4: Convert Probabilities to Percentages

Convert the probabilities to percentages:- Part (a): Approximately \( 70.74\% \) of goats weigh between 110 and 130 lb.- Part (b): Approximately \( 0.52\% \) of goats weigh less than 100 lb.- Part (c): Approximately \( 0.13\% \) of goats weigh more than 150 lb.

Key Concepts

Z-scores

A z-score measures how far and in what direction a data point deviates from the mean, measured in units of the standard deviation. The z-score formula is: \[ z = \frac{x - \mu}{\sigma} \] where:

• \(x\) is the value in question
• \(\mu\) is the mean of the distribution
• \(\sigma\) is the standard deviation

Z-scores allow us to compare scores on different scales or from different distributions. They tell us how unusual a value is within the context of its distribution. The z-score of 0 indicates that the value is exactly at the mean. Positive z-scores indicate values above the mean, while negative z-scores indicate values below the mean.
When we calculate z-scores, we are essentially standardizing our data, transforming it to fit the standard normal distribution. This transformation makes it easier to determine the probability of a value occurring within a given normal distribution.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are. It provides insight into the variability of the data relative to the mean. If the standard deviation is low, the numbers are close to the mean; if high, they are spread over a wider range.
Mathematically, it is defined by the square root of the variance. Variance itself is the average of the squared differences from the mean. For a data set, the formula for standard deviation \( \sigma \) is: \[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \] where:

• \(x_i\) represents each value in the data set,
• \(\mu\) is the mean of the data set,
• \(N\) is the number of data points.

Standard deviation is crucial in interpreting z-scores and understanding normal distributions. Knowing the standard deviation, we can determine how much of the data falls within certain ranges away from the mean.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a particular value. In the context of the standard normal distribution, it represents the area under the curve from the far left up to a specific z-score.
To find cumulative probabilities, we often use the standard normal distribution table or technology that provides these values quickly. These tables give us the probability that a z-score is less than a specified value. For example, a cumulative probability corresponding to a z-score of 1.5 or \( P(z < 1.5) \) would indicate the probability that a data point is less than 1.5 standard deviations above the mean.
In our analysis of normally distributed data, calculating cumulative probabilities helps determine the proportion of samples within certain limits. For example, if you want to know what percentage of data falls below or above a certain value, cumulative probability provides the answer. Ensure you understand how to read these tables and interpret their values as they are essential tools for evaluating normal distributions.