Question

A survey of enrollment at $35$community colleges across the United States yielded the following figures:

$$\begin{gathered} 6414;1550;2109;9350;21828;4300;5944;5722;2825; \\ 2044;5481;5200;5853;2750;10012;6357;27000;9414; \\ 7681;\text{3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493;} \\ \text{2771; 2861; 1263; 7285; 28165; 5080;}11622 \end{gathered}$$

a. Organize the data into a chart with five intervals of equal width. Label the two columns "Enrollment" and "Frequency."

b. Construct a histogram of the data.

c. If you were to build a new community college, which piece of information would be more valuable: the mode or the mean?

d. Calculate the sample mean.

e. Calculate the sample standard deviation.

f. A school with an enrollment of $8000$would be how many standard deviations away from the mean?

Short answer

(a)

Enrollment	Frequency
$0-6000$	$16$
$6000-12000$	$12$
$12000-18000$	$3$
$18000-24000$	$2$
$24000-30000$	$2$
Total	$35$

(b)

(c) The mode.

(d) The sample mean$=8485.71$

(e) The sample standard deviation$=6883$

(f) The value $8000$is $0.07$standard deviations away from the mean.

Step-by-step solution

Given information (part a)

The list of figures

$$\begin{gathered} 6414;1550;2109;9350;21828;4300;5944;5722;2825; \\ 2044;5481;5200;5853;2750;10012;6357;27000;9414; \\ 7681;\text{3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493;} \\ \text{2771; 2861; 1263; 7285; 28165; 5080;}11622 \end{gathered}$$

Explanation (part a)

Enrollment	Frequency
$0-6000$	$16$
$6000-12000$	12
$12000-18000$	3
$18000-24000$	$2$
$24000-30000$	$2$
Total	$35$

Given information (part b)

The list of figures

$$\begin{gathered} 6414;1550;2109;9350;21828;4300;5944;5722;2825; \\ 2044;5481;5200;5853;2750;10012;6357;27000;9414; \\ 7681;\text{3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493;} \\ \text{2771; 2861; 1263; 7285; 28165; 5080;}11622 \end{gathered}$$

Explanation (part b)

Given information (part c)

The list of figures

$$\begin{gathered} 6414;1550;2109;9350;21828;4300;5944;5722;2825; \\ 2044;5481;5200;5853;2750;10012;6357;27000;9414; \\ 7681;\text{3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493;} \\ \text{2771; 2861; 1263; 7285; 28165; 5080;}11622 \end{gathered}$$

Explanation (part c)

The mean is the average value of a given data set which represents each and every observation. The mean is dependent on the outliners too. But the mode is the observation that gives the highest frequency. when we have to make a decision to build a new community college, the best way is to consider the majority. Hence, in this case, the mode is more valuable.

Give information (part d)

The list of figures

$$\begin{gathered} 6414;1550;2109;9350;21828;4300;5944;5722;2825; \\ 2044;5481;5200;5853;2750;10012;6357;27000;9414; \\ 7681;\text{3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493;} \\ \text{2771; 2861; 1263; 7285; 28165; 5080;}11622 \end{gathered}$$

Explanation (part d)

Enrollment	Frequency (f)	Class mark (x)	fx
$0-6000$	$16$	$3000$	$48000$
$6000-12000$	$12$	$9000$	$108000$
$12000-18000$	$3$	$15000$	$45000$
$18000-24000$	$2$	$21000$	$42000$
$24000-30000$	$2$	$27000$	$54000$
Total	$35$		$297000$

$$\begin{gathered} \overline{x}=\frac{\sum fx}{\sum f} \\ =\frac{297000}{35} \\ =8485.7143 \end{gathered}$$

Given information (part e)

The list of figures

$$\begin{gathered} 6414;1550;2109;9350;21828;4300;5944;5722;2825; \\ 2044;5481;5200;5853;2750;10012;6357;27000;9414; \\ 7681;\text{3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493;} \\ \text{2771; 2861; 1263; 7285; 28165; 5080;}11622 \end{gathered}$$

Step 10: 

Class mark (x)	Frequency (f)	$(x-\overline{x}{)}^2$	$f(x-\overline{x}{)}^2$
$3000$	$16$	$30096196$	$481539136$
$9000$	$12$	$264196$	$3170352$
$15000$	$3$	$42432196$	$127296588$
$21000$	$2$	$156600196$	$313200392$
$27000$	$2$	$342768196$	$685536392$
Total	$35$		$1610742860$

$$\begin{gathered} S=\sqrt{\frac{\sum f(x-\overline{x}{)}^2}{\sum f-1}} \\ =\sqrt{\frac{1610742860}{35-1}} \\ =\sqrt{\frac{1610742860}{34}} \\ =\sqrt{47374790} \\ =6882.9347 \end{gathered}$$

Given information (part f)

The list of figures

$$\begin{gathered} 6414;1550;2109;9350;21828;4300;5944;5722;2825; \\ 2044;5481;5200;5853;2750;10012;6357;27000;9414; \\ 7681;\text{3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493;} \\ \text{2771; 2861; 1263; 7285; 28165; 5080;}11622 \end{gathered}$$

Sample mean$\overline{x}=8486$

The sample standard deviation$S=6883$

Value to be considered$x=8000$

Explanation (part f)

$$\begin{gathered} (\#\text{of STDVs})=\frac{\text{value-}\overline{x}}{S} \\ (\#\text{of STDVs})=\frac{8000-8486}{6883} \\ =\frac{-486}{6883} \\ =-0.0706 \end{gathered}$$

Hence $8000$is $0.07$standard deviation below the mean.