Writing Comparisons Between Two Population Parameters

Learning Outcomes

  • Express verbal comparisons of two population parameters as mathematical expressions

Sometimes we are interested in comparing two population parameters. For example, a researcher may want to test whether a drug is effective in lowering blood pressure. Participants in a drug trial are randomly assigned to treatment and control groups. The treatment group will take the new drug for six weeks, while the control group will receive standard care. The blood pressures of participants will be recorded at the end of six weeks and the means for the two groups will be compared.

The researcher’s initial hypothesis might be stated: the mean blood pressure for the treatment group is lower than the mean blood pressure for the control group. In symbols, we might write

[latex]\large\mu _\mathrm{treatment} < \mu _\mathrm{control}[/latex].

We previously saw the connection between verbal statements about the relationship between a variable x and a constant k and the corresponding mathematical statement.

Verbal statement
x is …		 Mathematical Statement
greater than or equal to k
at least k
not less than k		 [latex]x \geq k[/latex]
less than or equal to k
at most k
not more than k		 [latex]x \leq k[/latex]
less than k
below k
fewer than k		 [latex]x < k[/latex]
greater than k
more than k
above k		 [latex]x > k[/latex]
equal to k
is k
exactly k
the same as k		 [latex]x=k[/latex]
not equal to k
not k
different from k		 [latex]x \neq k[/latex]

We can apply the same principles to comparisons involving two population parameters.

Example

Verbal Statement Mathematical Statement
Proportion A is at least proportion B. [latex]p_A \geq p_B[/latex]
Mean 1 is no greater than mean 2. [latex]\mu _1 \leq \mu _2[/latex]
The average time for dogs and cats is the same. [latex]\mu _\mathrm{dogs} = \mu _\mathrm{cats}[/latex]
The mean for college X is more than the mean for college Y. [latex]\mu _X > \mu _Y[/latex]

Example

Write the following statement as a comparison between two population parameters.

A researcher states the mean sodium content of soup from manufacturer A is no more than the mean sodium content of soup from manufacturer B.

Example

Write the following statement as a comparison between two population parameters.

A marketing researcher states the percentage of Michigan residents that regularly watch professional football is 5 percentage points lower than the percentage of Wisconsin residents who regularly watch professional football.

Example

Write the following statement as a comparison between two population parameters.

A physical therapist suggests that the average recovery time for patients is lower for those patients receiving massage therapy in addition to standard treatment as compared to patients receiving standard treatment alone.