# Which samples result in a sample mean that overestimates the population mean...?

## "Pulse Rates" • The following data represent the pulse rates (beats per minute) of nine students enrolled in a section of Sullivan’s Introductory Statistics course. Treat the nine students as a population. Student Pulse Perpectual Bempah 76 Megan Brooks 60 Jeff Honeycutt 60 Clarice Jefferson 81 Crystal Kurtenbach 72 Janette Lantka 80 Kevin McCarthy 80 Tammy Ohm 68 Kathy Wojdyla 73 Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Jul 2, 2017

(1) Try picking the last six students and taking the sample mean here.
(2) You could take the first five students when taking the sample mean here.
(3) Here, you need to get close to $72. \overline{22}$, so maybe Tammy, Crystal, Kathy, and Perpectual. You would then get $72.25$, which is pretty close...

The remainder of this answer expands on the first question.

This is asking you to compare the population mean (the average based on the entire selection of students), versus the sample mean (the average based on a fraction of this selection) of the heart rates of each student.

Each type of mean is defined in the same way, except for the number of students chosen:

$\overline{x} = {\sum}_{i = 1}^{N} {x}_{i} / N$

where:

• $\overline{x}$ is the mean (either sample or population).
• $i$ is the index for each student.
• $N$ is the total number of students.
• ${x}_{i}$ is the data point corresponding to each student.

The population was stated to be all 9 students ("Treat the nine students as a population"). The sample can be any fraction of these students you want.

You are free to pick a collection of these 9 students to estimate what the population mean is. This is called the sample of the population, and is useful if you don't want to take data on so many people that it's unmanageable.

The population mean is:

${\overline{x}}_{\text{pop}} = \frac{76 + 60 + 60 + 81 + 72 + 80 + 80 + 68 + 73}{9} = 72. \overline{22}$

For the first question, you are supposed to pick a sample, a fraction, of these 9 students, that gives a mean above $72. \overline{22}$.

Any student whose heart rate is above $72$ (such as Clarice, Janette, or Kevin) will contribute to an overestimate when taking the mean, because the mean is weighted the most in one direction by the numbers that are farthest from the mean in that direction.

So, those students with a heart rate of $73$ and above (in this case, anyway) are good candidates.

One choice for a sample mean is the last six students:

${\overline{x}}_{\text{smp}} = \frac{81 + 72 + 80 + 80 + 68 + 73}{6} = \textcolor{b l u e}{75. \overline{66} > 72. \overline{22}}$

So, this choice for a sample generates a sample mean that overestimates the population mean, i.e. ${\overline{x}}_{\text{smp" > barx_"pop}}$.