• Describe a population and sample of interest.
• Use technology to find sample statistics from a sampling distribution.

parameters & statistics

We call the characteristic values of a sample statistics.

 

Characteristic values of a population are called parameters.

 

Remember:

• A statistic describes a characteristic of a sample.
• A parameter describes a characteristic of a population.

Consider the following statistical question: “What is the average amount of money spent on textbooks per semester by full-time students in the three Arizona public universities?”

We cannot interview every full-time student in all of the three Arizona public universities because it would take too much time and cost too much money. We therefore carefully select a sample of full-time students from all three Arizona public universities to represent the population of all full-time students. Then we collect data from the sample to estimate the average amount spent on textbooks.

A well-stated research question contains information about:

• The population: Full-time students from all three Arizona public universities.
• The information we will collect from each individual in the sample. We call this the variable. The variable is what we plan to measure  and is often represented by [latex]X[/latex]: In our case, this is the amount of money spent on textbooks per semester.
• A numerical characteristic about the population related to this variable: The average amount of money spent on textbooks per semester.

A numerical characteristic about a population is called a parameter. In the example above we are interested in the average, or mean, amount of money all students spent on textbooks per semester.

We use information from the sample to estimate the population mean. A numerical characteristic of a sample is called a sample statistic.

Another research question might be whether the majority of students qualify for federal student loans.

• The population: Full-time students from all three Arizona public university.
• The information we will collect from each individual in the sample is the variable: Whether or not each student in the sample qualifies for federal student loans.
• The related population’s numerical characteristic: The proportion of all students from all three Arizona public universities that qualify for student loans

The parameter is the population proportion and the sample statistic is the sample proportion.

If [latex]3[/latex] out of [latex]5[/latex] people in our sample agree with a statement, this corresponds to the sample proportion [latex]\hat{p}=\frac{3}{5}=0.6[/latex].

If information includes all members of the population, the information refers to a population parameter. The word “every” or “each” or “all” in a statement can be a clue that an entire population has been included.

A numerical characteristic refers to a sample statistic if only a portion of the population has been included. Information likely refers to a sample if a statement refers to a relatively small sample of a large population. 

Determine whether the number refers to a population parameter or a sample statistic.

1. [latex]78\%[/latex] of [latex]50[/latex] athletes at a large university said they were happy they chose to continue competing at the collegiate level.
2. [latex]706[/latex] of the [latex]2,223[/latex] passengers on the Titanic survived.

Show answer

1. Since a large university likely has many more than [latex]50[/latex] athletes, and the statement refers to a survey, [latex]78\%[/latex] refers to a sample statistic, [latex]\hat{p}[/latex].
2. The statement refers to all passengers on the Titanic, so [latex]706[/latex] out of [latex]2,223[/latex], or [latex]31.8\%[/latex], refers to a population proportion, [latex]p[/latex].