Question

In Exercise 17 we suggested the model \(N(1152,84)\) for weights in pounds of yearling Angus steers. What weight would you consider to be unusually low for such an animal? Explain.

Short answer

A weight below 984 pounds is unusually low for yearling Angus steers.

Step-by-step solution

Understanding the Problem

We're given a normal distribution model for the weights of yearling Angus steers, specifically, normally distributed with a mean (\(\mu\)) of 1152 pounds and a standard deviation (\(\sigma\)) of 84 pounds. The task is to find a weight that would be considered unusually low.

Identifying Unusual Values

In statistics, values that fall outside 2 standard deviations from the mean are typically considered unusual. We will calculate 2 standard deviations below the mean to find the unusually low weight.

Calculating Lower Boundary

We calculate the lower boundary using the formula: \[x = \mu - 2\sigma\]Substituting the given values, we have \[1152 - 2 \times 84 = 1152 - 168 = 984\]Thus, any weight below 984 pounds is considered unusually low.

Key concepts

Standard Deviation

Standard deviation is a fundamental concept in statistics, representing the amount of variability or spread in a dataset. When we have a set of data points, some of them might deviate from the average, or mean, value more than others. This is where the standard deviation comes in. It gives us an average distance that these data points lie from the mean.

For example, in the normal distribution mentioned in the exercise, the weights of yearling Angus steers have a standard deviation of 84 pounds. This number quantifies how much the weights vary from the mean weight of 1152 pounds.

• Low standard deviation: Data points are close to the mean.
• High standard deviation: Data spreads over a wide range of values.

Standard deviation helps in understanding if the data is tightly clustered around the mean or more dispersed. When finding usual or unusual values, like an unusually low weight, standard deviation assists in setting the boundaries by indicating the typical spread.

Mean Value

The mean value, often simply referred to as the 'average' or \(\mu\), is a measure of central tendency in a data set. It represents the central point of a set of numbers. You calculate the mean by adding up all the values and dividing by their count. In the context of a normal distribution, it is the peak of the bell curve where most data points tend to cluster.

For the problem at hand, the mean weight of yearling Angus steers is given as 1152 pounds. This is the expected weight around which most of the steers will weigh.

• Mean helps in summarizing a data set with a single value.
• It is sensitive to the specific values and can shift with outliers.

When analyzing data through statistical modeling, the mean helps provide a baseline or reference point. In practical applications, knowing whether a specific data point is unusually high or low often depends first on comparing it to the mean value.

Statistical Modeling

Statistical modeling involves creating mathematical representations of real-world processes. These models help predict and understand variability in data. A normal distribution model, like the one used for the Angus steers, is a type of statistical model that assumes data points are dispersed in a characteristic bell-shaped curve around the mean.

Using statistical models, we can deduce patterns and make inferences about large data populations.

• They offer predictions based on sampled data.
• Statistical models can identify and label data as expected or unusual.

In the exercise, the mean and standard deviation are employed to construct a normal distribution model. It helps us identify what is considered a typical or atypical weight for this population of steers. Essentially, statistical modeling allows us to make educated judgments from data, enhancing decision-making processes.