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

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"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?

"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?

##### 1 Answer

(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

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

**mean (the average based on a fraction of this selection) of the heart rates of each student.**

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

#barx = sum_(i=1)^(N) x_i/N# where:

#barx# 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:

#barx_"pop" = (76 + 60 + 60 + 81 + 72 + 80 + 80 + 68 + 73)/9 = 72.bar(22)#

For the first question, you are supposed to pick a sample, a fraction, of these 9 students, that gives a mean *above*

Any student whose heart rate is **above** ** most** in one direction by the numbers that are

**from the mean in that direction.**

*farthest*So, those students with a heart rate of

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

#barx_"smp" = (81 + 72 + 80 + 80 + 68 + 73)/6 = color(blue)(75.bar(66) > 72.bar(22))#

*So, this choice for a sample generates a sample mean that overestimates the population mean, i.e.*

*.*