Question

The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1152 pounds. Suppose that weights of all such animals can be described by a Normal model with a standard deviation of 84 pounds. a) How many standard deviations from the mean would a steer weighing 1000 pounds be? b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds?

Short answer

It's more unusual for a steer to weigh 1000 pounds than 1250 pounds.

Step-by-step solution

Understanding the Z-score

To find how many standard deviations a given weight is from the mean, we calculate the Z-score using the formula:\[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the weight, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Calculate Z-score for 1000 pounds

Substitute \( X = 1000 \), \( \mu = 1152 \), and \( \sigma = 84 \) into the Z-score formula:\[ Z_{1000} = \frac{1000 - 1152}{84} = \frac{-152}{84} \approx -1.81 \]This means that a steer weighing 1000 pounds is approximately \(-1.81\) standard deviations from the mean.

Calculate Z-score for 1250 pounds

Substitute \( X = 1250 \), \( \mu = 1152 \), and \( \sigma = 84 \) into the Z-score formula:\[ Z_{1250} = \frac{1250 - 1152}{84} = \frac{98}{84} \approx 1.17 \]This means that a steer weighing 1250 pounds is approximately \(1.17\) standard deviations from the mean.

Compare Z-scores to Determine Unusualness

The further the Z-score is from 0, the more unusual the value is in a normal distribution. A Z-score of \(-1.81\) is farther from 0 than \(1.17\), meaning a steer weighing 1000 pounds is more unusual than one weighing 1250 pounds.

Key concepts

Standard Deviation

Standard deviation is a fundamental statistical tool that helps us measure the amount of variation or dispersion within a dataset. Imagine you have several weights of yearling Angus steers, to understand how spread out these weights are, you use standard deviation. In the given exercise, the standard deviation was 84 pounds.

A larger standard deviation indicates that the data points are more spread out from the mean. Conversely, a smaller standard deviation means they are closer to the mean average. This concept is essential when talking about normal distribution. It allows us to see just how much individual values deviate from the average value. In this case, how much each steer's weight deviates from the average 1152 pounds. Understanding this gives us insight into how typical or atypical any given steer’s weight is.

Z-score

The Z-score is a value that tells us how many standard deviations a data point is from the mean. Calculating a Z-score is essential for understanding the position of a data point within a normal distribution, which helps in determining how unusual or usual the data point is.

The formula used for calculating Z-score is: \[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the value (weight of the steer),
• \(\mu\) is the mean, and
• \(\sigma\) is the standard deviation.

In the exercise, the Z-score for a 1000-pound steer was \(-1.81\). This means the steer weighs 1.81 standard deviations less than the mean weight of 1152 pounds. Similarly, the Z-score for a steer weighing 1250 pounds was calculated to be \(1.17\), indicating it is 1.17 standard deviations heavier than the mean.

Mean

In any dataset, the mean is an essential central tendency measure. It is calculated by summing all the values and then dividing by the number of values; it gives you the average.

In the case of yearling Angus steers, the mean weight is given as 1152 pounds. This number essentially tells us that if you took the weight of all steers and averaged them, each would weigh around 1152 pounds.

• The mean provides a benchmark for comparison. Without it, we'd have no way of knowing what's considered light or heavy.
• It aids significantly in calculating other important metrics like Z-scores and reveals how individual data compares to the overall group.

You can think of the mean as the point of balance in the dataset where every data point contributes its own little weight to the determination.

Unusualness in Statistics

Unusualness in statistics relates to how far a data point lies from the average or mean. In a normally distributed dataset, values near to the mean are considered common, while values much greater or less than the mean are deemed unusual.

The degree of unusualness often depends on how many standard deviations the data point is from the mean. In a normal distribution:

• Approximately 68% of data falls within one standard deviation.
• 95% falls within two standard deviations, and
• 99.7% lies within three standard deviations.

So, considering the Z-scores calculated in the exercise, a steer weighing 1000 pounds was \(-1.81\) standard deviations from the mean and was more unusual compared to the one weighing 1250 pounds, which was only \(1.17\) standard deviations from the mean. This indicates that weights significantly less or more than the mean — when Z-scores are respectively negative or positive and far from zero — are more unusual.