Question

Recall that the beef cattle described in Exercise 17 had a mean weight of 1152 pounds, with a standard deviation of 84 pounds. a) Cattle buyers hope that yearling Angus steers will weigh at least 1000 pounds. To see how much over (or under) that goal the cattle are, we could subtract 1000 pounds from all the weights. What would the new mean and standard deviation be? b) Suppose such cattle sell at auction for 40 cents a pound. Find the mean and standard deviation of the sale prices for all the steers.

Short answer

a) New Mean: 152 pounds, New SD: 84 pounds; b) Mean Sale Price: $460.80, SD: $33.60.

Step-by-step solution

Understanding the Problem

We have a set of beef cattle weights that are Normally distributed with a mean (\( \mu \)) of 1152 pounds and a standard deviation (\( \sigma \)) of 84 pounds. We need to calculate new statistics based on adjustments to these weights.

Part a: Calcuating Adjusted Mean and Standard Deviation

Subtracting 1000 pounds from each weight does not change the standard deviation. The new mean becomes the original mean minus 1000, so:\[ \text{New Mean} = 1152 - 1000 = 152 \] And the standard deviation remains:\[ \text{New Standard Deviation} = 84 \]

Part b: Calculating Sale Prices Mean and Standard Deviation

Sale prices are 40 cents per pound, meaning each weight is multiplied by 0.40. This scales both the mean and the standard deviation by 0.40:\[ \text{Mean Sale Price} = 0.40 \times 1152 = 460.8 \] \[ \text{Standard Deviation of Sale Price} = 0.40 \times 84 = 33.6 \]

Key concepts

Mean

The mean, often referred to as the average, is a fundamental concept in descriptive statistics. It provides a central value that represents a set of data. To calculate the mean, you simply add up all the figures in your data set and then divide by the total number of values.

For instance, in the context of beef cattle weights mentioned in the exercise, the mean weight of the cattle is given as 1152 pounds. This number signifies the average weight of all the cattle in the dataset.

When adjustments are made to the dataset, such as subtracting a fixed number from each data value, the mean will adjust accordingly. For example, if every cattle's weight is reduced by 1000 pounds, the mean also decreases by 1000 pounds. So, the new mean would be 1152 - 1000 = 152 pounds.

It's important to remember that while changing each value in your dataset by the same amount will affect the mean, it does not impact the spread or variability of the dataset.

Standard Deviation

Standard deviation is a measure of variability or spread in a set of data. It tells us how much individual data points differ from the mean of the dataset.

In practical terms, a low standard deviation means the data points tend to be close to the mean, while a high standard deviation indicates the data are spread out over a wider range of values.

In the cattle weight example, the standard deviation is 84 pounds. This tells us that the weights of the cattle vary, on average, by 84 pounds from the mean weight of 1152 pounds.

An interesting property of standard deviation is that it remains unchanged when each data point is increased or decreased by the same amount. However, it does change if each value is multiplied by or divided by a constant. This is why when subtracting a number (e.g., 1000 pounds) from each weight, the standard deviation remains at 84 pounds. When the weights are multiplied by 0.40 to find sale prices, the standard deviation is scaled by the same factor, resulting in a new standard deviation of 33.6 for sale prices.

Normal Distribution

The normal distribution is an important concept in statistics and is often referred to as a "bell curve" due to its distinct shape. It describes a distribution where most values cluster around a central point (mean) and the probabilities of values taper off equally in both directions from the mean.

The normal distribution is symmetric and its shape is defined by its mean and standard deviation.

In the context of the beef cattle weight problem, the weights follow a normal distribution with a mean of 1152 pounds and a standard deviation of 84 pounds. This tells us that most cattle weights are centered around 1152 pounds and the probability of extreme weights (either very low or very high) decreases as we move away from the mean.

Understanding normal distribution helps in making predictions about the dataset. For example, using the properties of this distribution, one can determine the percentage of cattle that fall within certain weight ranges. Furthermore, transformations like subtracting a constant from all data points or scaling can be understood within the context of their impact on a normally distributed dataset.